tag:blogger.com,1999:blog-18911570766354416512024-03-18T04:01:18.904+01:00Matière et temps. Réalité Générale de l'Univers. R.G.U.L'univers est nombre.
Matière et temps.
Une nouvelle approche de l'arbre des connaissances.
Réalité Générale de l'Univers.
R.G.U.alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.comBlogger578125tag:blogger.com,1999:blog-1891157076635441651.post-68278772631430753512049-06-01T10:31:00.001+02:002023-02-05T16:16:55.190+01:00Suites de CONWAY<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-7eDWUnLqT8Q/TxmRtgyJ6nI/AAAAAAAAo7o/lrrDDFu2yno/s1600/allemagne+klein.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="247" src="https://3.bp.blogspot.com/-7eDWUnLqT8Q/TxmRtgyJ6nI/AAAAAAAAo7o/lrrDDFu2yno/s400/allemagne+klein.jpg" width="400" /></a></div>
<br />
<br />
<br />
bien intéressant,<br />
<br />
le nombre traduit une évolution de contenu temporel.<br />
<br />
Il y a une base de départ. écriture à gauche.<br />
<br />
0 inexistence<br />
1 existence<br />
<br />
Il faut traduire ce contenu spatial (matériel) et temporel (immatériel) de 0 et 1.<br />
A l'image de Conway, écriture gauche droite ou droite gauche :<br />
<br />
avec ajout du 1:<br />
0 >>>>> 10 ou 01<br />
1 >>>>> 11<br />
<br />
avec ajout du 0 :<br />
0>>>>> 00<br />
1 >>>>>01 ou 10<br />
<br />
puis on itère :<br />
10 >>>> 110 ou 101<br />
01 >>>>> 101 ou 011<br />
11 >>>>> 111<br />
00 >>>>>000<br />
01 >>> 001 ou 010<br />
<br />
.....<br />
<br />
<br />
<br />
<br />
0<br />
10<br />
1011<br />
211011<br />
21102112<br />
122112102112<br />
<br />
....<br />
<br />
1<br />
11<br />
21<br />
<br />
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
<br /></div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
Terme 4 = 1211</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
Terme 5 = 111221</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
Terme 6 = 312211</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
Terme 7 = 13112221</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
<br /></div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
....</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
11</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
21</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
<br /></div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
.....</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
<br /></div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
01</div>
<div style="background-color: white; font-family: sans-serif; font-size: 13px; line-height: 19px; margin-bottom: 0.5em; margin-top: 0.4em;">
1011</div>
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<br />
<a href="http://eljjdx.canalblog.com/archives/2014/01/19/28956887.html">http://eljjdx.canalblog.com/archives/2014/01/19/28956887.html</a><br />
<a href="http://oeis.org/A014967">http://oeis.org/A014967</a> <br />
<br />
<a href="http://mathworld.wolfram.com/CosmologicalTheorem.html">http://mathworld.wolfram.com/CosmologicalTheorem.html</a> <br />
<br />
<a href="http://mathworld.wolfram.com/LookandSaySequence.html">http://mathworld.wolfram.com/LookandSaySequence.html</a> <br />
<br />
<a href="http://mathworld.wolfram.com/ConwaysConstant.html">http://mathworld.wolfram.com/ConwaysConstant.html</a> <br />
<br />
<a href="http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/horton.html">http://www.math.rutgers.edu/~zeilberg/mamarim/mamarimhtml/horton.html</a> <br />
<br />
<a href="http://www.cs.cmu.edu/~kw/pubs/conway.pdf">http://www.cs.cmu.edu/~kw/pubs/conway.pdf</a><br />
<br />
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<a href="http://2.bp.blogspot.com/-eclOW3UyhS0/TxmR0K5LaEI/AAAAAAAAo7w/6kTYdHjlfo0/s1600/allemagne+klein1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="247" src="https://2.bp.blogspot.com/-eclOW3UyhS0/TxmR0K5LaEI/AAAAAAAAo7w/6kTYdHjlfo0/s400/allemagne+klein1.jpg" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><div style="background-color: white; color: #222222; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: 13.2px; text-align: start;"><span style="color: #ff00fe; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; font-size: 13px;"><i><a href="http://fr.wikipedia.org/wiki/Charles_Hermite" style="color: #cc6611; text-decoration-line: none;" target="_blank">Charles Hermite</a> :</i></b><br style="color: black; font-weight: 400; text-align: center;" /><b style="color: red; font-family: Arial, Helvetica, sans-serif;"><i><br />« Je vous ferais bondir,</i></b></span></b></span></div><div style="background-color: white; text-align: start;"><div style="text-align: start;"><span style="color: red; font-family: Arial, Helvetica, sans-serif;"><b><i><br /></i></b></span></div><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><span style="color: black; font-weight: 400; text-align: center;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; text-align: start;"><i> si j'osais vous avouer que je n'admets aucune solution de continuité, </i></b></span></span></b></span></div><div style="background-color: white; text-align: start;"><div style="text-align: start;"><span style="color: red; font-family: Arial, Helvetica, sans-serif;"><b><i><br /></i></b></span></div><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><span style="color: black; font-weight: 400; text-align: center;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; text-align: start;"><i>aucune coupure entre les mathématiques et la physique,</i></b></span></span></b></span></div><div style="background-color: white; text-align: start;"><div style="text-align: start;"><br /></div><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><span style="color: black; font-weight: 400; text-align: center;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; text-align: start;"><i> et que les nombres entiers me semblent exister en dehors de nous et en s'imposant</i></b></span></span></b></span></div><div style="background-color: white; text-align: start;"><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><span style="color: black; font-weight: 400; text-align: center;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; text-align: start;"><i><br /></i></b></span></span></b></span></div><div style="background-color: white; text-align: start;"><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><span style="color: black; font-weight: 400; text-align: center;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; text-align: start;"><i> avec la même nécessité, la même fatalité que le sodium, le potassium, etc. »</i></b><br /></span><span style="color: black; font-weight: 400; text-align: center;"></span><b style="color: red; font-family: Arial, Helvetica, sans-serif; font-size: 13px;"><i><br /></i></b><span style="color: black; font-weight: 400; text-align: center;"></span><b style="color: red; font-family: Arial, Helvetica, sans-serif; font-size: 13px;"><i><br />— Correspondance avec Stieltjes, janv. 1889, Paris, éd. Gauthier-Villars, 1905, t. I, p. 332</i></b></span></b></span></div><div style="background-color: white; text-align: start;"><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; font-size: 13px;"><i><br /></i></b></span></b></span></div><div style="background-color: white; text-align: start;"><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; font-size: 13px;"><i><br /></i></b></span></b></span></div><div style="background-color: white; text-align: start;"><span style="color: #ff00fe; font-family: Arial, Tahoma, Helvetica, FreeSans, sans-serif; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; font-size: 13px;"><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge5iSuqXN-yIzkU39ZPCL4mxlrezgxvhUnEAR2MMOhfoM8eOxKqK2HB9CKeWNjp4uoRiTd2c-9LfRSfDGUp8CFXc0-p-MsCRnIIfaD7manKkvyMEInr--uLb_VKqfQJnglPZOoFGGsDCh-CZmXMFAW09am65ZaIOsEyU6lWiZgvjYnsRDTe5ro-C5E/s788/Charles_Hermite_circa_1887.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="788" data-original-width="500" height="511" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEge5iSuqXN-yIzkU39ZPCL4mxlrezgxvhUnEAR2MMOhfoM8eOxKqK2HB9CKeWNjp4uoRiTd2c-9LfRSfDGUp8CFXc0-p-MsCRnIIfaD7manKkvyMEInr--uLb_VKqfQJnglPZOoFGGsDCh-CZmXMFAW09am65ZaIOsEyU6lWiZgvjYnsRDTe5ro-C5E/w324-h511/Charles_Hermite_circa_1887.jpg" width="324" /></a></div><br /><i><br /></i></b></span></b></span></div><div><span style="color: #ff00fe; font-size: x-large;"><b style="color: black; font-size: medium;"><span style="color: #351c75;"><b style="color: red; font-family: Arial, Helvetica, sans-serif; font-size: 13px;"><i><br /></i></b></span></b></span></div></div>
alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-73752559924756564972025-08-18T16:32:00.000+02:002016-09-01T20:15:09.064+02:00Un nombre "homogène" fondamental<div class="separator" style="clear: both; text-align: center;">
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<br />
<span style="background-color: yellow;">période du développement décimal 60</span><br />
<br />
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<a href="http://2.bp.blogspot.com/--n-0DfdDVUc/VdM-_X-kPHI/AAAAAAABDbU/TDpwfA1Ycbo/s1600/homog%25C3%25A8ne1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://2.bp.blogspot.com/--n-0DfdDVUc/VdM-_X-kPHI/AAAAAAABDbU/TDpwfA1Ycbo/s640/homog%25C3%25A8ne1.jpg" width="616" /></a></div>
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<a href="http://1.bp.blogspot.com/-fjs20tng1Pg/VdM_TRnbZlI/AAAAAAABDbk/4D7Fwl-Xxv0/s1600/homog%25C3%25A8ne.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="562" src="https://1.bp.blogspot.com/-fjs20tng1Pg/VdM_TRnbZlI/AAAAAAABDbk/4D7Fwl-Xxv0/s640/homog%25C3%25A8ne.jpg" width="640" /></a></div>
il suffit d'ajouter un zéro à droite pour avoir autant de chacun des chiffres ......<br />
ou de considérer le 0 de gauche de 1/61 après la virgule, significatif 0.<span style="color: red; font-size: large;">0</span>1639344...<br />
<br />
60 secondes 60 minutes 360 ° 24 heures<br />
<br />
La genèse et la création du monde en 6 jours<br />
<br />
6^10 = 604<span style="color: red;"> 66</span> 176 ; 604+176 =780<br />
<br />
<strong><span style="background-color: yellow; font-size: x-large;">Et 10! secondes = 6 semaines</span></strong><br />
<strong><span style="background-color: yellow; font-size: x-large;"><br /></span></strong>
<strong><span style="background-color: yellow; font-size: x-large;">Voir x/61</span></strong><br />
<br />
<a href="https://fr.wikipedia.org/wiki/D%C3%A9veloppement_d%C3%A9cimal_p%C3%A9riodique_de_l%27inverse_d%27un_nombre_premier">https://fr.wikipedia.org/wiki/D%C3%A9veloppement_d%C3%A9cimal_p%C3%A9riodique_de_l%27inverse_d%27un_nombre_premier</a>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-37349407406720089222025-01-10T18:04:00.002+01:002021-03-04T18:08:45.644+01:00Pi et ckplan volumes (Un moins epsilon)<div style="text-align: center;">
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<span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><a href="http://www.astrosurf.com/luxorion/cosmos-modelesunivers.htm" target="_blank">Source</a></span></span><br />
<span style="color: red;"><br /></span><br />
<span style="color: red;"><br /></span><br />
<br />
<br />
On retrouve par exemple <b>888</b>178 qui sont les premiers chiffres de 1/(2^50) ...<br />
<br />
puis <b>444</b>089 1/2^51<br />
puis <b>222</b>045 1/2^52<br />
<br />
<br />
<br />
<br />
<span style="color: red; font-size: large;"><strong></strong></span><br />
<span style="color: red; font-size: large;"><strong>777,999,666,888,555 </strong></span><br />
<br />
On passe par 2*pi ...<br />
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<span style="color: red;">1******</span><br /><div style="text-align: center;"><i style="color: #20124d; font-size: xx-large;"><span><b>Commentaire reçu (Anonyme) :</b></span></i></div>
<i><div style="text-align: center;"><span style="color: red;"><br /></span></div></i><i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b>De même la "remarquable" formule qui donne à peu près pi * 10^(-14).</b></span></span></i><br />
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b><br /></b></span></span></i>
<span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;">Non, si eps ,ci-après est égal à 10-7,on obtient 7 décimales de pi 1415926 ...</span></span><br />
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b><br /></b></span></span></i>
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b> Si vous développez 4/3 - 4/3*(1- 0.5 eps)^3 - 2*eps </b></span></span></i><br />
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b><br /></b></span></span></i>
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b>(où j'ai remplacé 10^-7 par eps) </b></span></span></i><br />
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b><br /></b></span></span></i>
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b>vous trouvez </b></span></span></i><br />
<i><span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;"><b>-eps^2 + 1/6 eps^3 = -(1-1/6*10^-7)*10^-14 : </b></span></span></i><br />
<i><span style="color: red;"><b><span face=""arial" , sans-serif" style="background-color: white;">votre résultat est exactement ce facteur multiplié par pi.</span></b></span></i><br />
<i><span style="color: red;"><b><br style="background-color: white; font-family: arial, sans-serif;" /><span face=""arial" , sans-serif" style="background-color: white;">C'est donc des "trivialités", et en allant au bout des petits calculs élémentaires lorsque vous faites une telle "découverte", </span></b></span></i><br />
<i><span style="color: red;"><b><span face=""arial" , sans-serif" style="background-color: white;"><br /></span></b></span></i>
<i><span style="color: red;"><b><span face=""arial" , sans-serif" style="background-color: white;">vous trouvez vite l'explication simple.</span></b></span></i><br />
<span style="color: red;"><span face=""arial" , sans-serif" style="background-color: white;">1******</span></span><br />
<span style="color: red;">3******</span><br />
<b><span style="font-size: large;"><span>Remarque : avec eps de la forme 5*10^-x ou 0.5*10^-x ....,</span><br />
<span><span face=""arial" , sans-serif" style="background-color: white;"></span></span><br />
<span>on n'obtient des décimales de pi qu'avec x > 2 ......</span><br /></span></b>
<span style="color: red;"><br /></span>
<span style="color: red;">3.*****</span><br />
ckplan <b>555</b>382 .....<br />
<br />alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com1tag:blogger.com,1999:blog-1891157076635441651.post-9780091885828651262024-12-31T08:22:00.001+01:002021-02-18T10:46:58.084+01:00Melancolia Dürer<br />
<a href="http://fr.wikipedia.org/wiki/Melencolia_de_D%C3%BCrer">http://fr.wikipedia.org/wiki/Melencolia_de_D%C3%BCrer</a><br />
<br />
<a href="http://commons.wikimedia.org/wiki/File%3AD%C3%BCrer_Melancholia_I.jpg" title="Albrecht Dürer [Public domain], via Wikimedia Commons"><img alt="Dürer Melancholia I" height="640" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/18/D%C3%BCrer_Melancholia_I.jpg/256px-D%C3%BCrer_Melancholia_I.jpg" width="504" /></a><br />
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<div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;">
Sur le mur derrière l'ange, figure un <a href="https://fr.wikipedia.org/wiki/Carr%C3%A9_magique_(math%C3%A9matiques)" style="background: none; color: #0b0080; text-decoration-line: none;" title="Carré magique (mathématiques)">carré magique</a>, dont la valeur est 34. Les carrés magiques sont, notamment dans les ésotérismes juif et islamique, associés à des connaissances secrètes qui furent transmises, pendant et avant l'époque de Dürer par des confréries d'ésotérisme chrétien qui maintenaient des relations suivies avec les initiés à l'ésotérisme islamique.</div>
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En ordonnant les nombres de 1 à 16 (ou à 9, 25 ou tout autre <a class="mw-redirect" href="https://fr.wikipedia.org/wiki/Nombre_carr%C3%A9" style="background: none; color: #0b0080; text-decoration-line: none;" title="Nombre carré">nombre carré</a> supérieur à 4), une grille carrée peut être remplie de façon telle que la somme sur chaque ligne horizontale, verticale ou diagonale ait la même valeur. Les carrés magiques utilisés dans l'hermétisme sont d'ordre <i>n</i>, c'est-à-dire qu'ils ont <i>n</i> lignes et <i>n</i> colonnes, correspondant aux entiers allant de 1 à <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle n^{2}}</annotation></semantics></math></span><img alt="n^{2}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ac9810bbdafe4a6a8061338db0f74e25b7952620" style="border: 0px; display: inline-block; height: 2.676ex; margin: 0px; vertical-align: -0.338ex; width: 2.467ex;" /></span>. La <i>somme de tous les nombres</i> d'un tel carré magique de taille <i>n</i> a pour valeur :</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 1+2+\cdots +n^{2}={\frac {n^{2}(n^{2}+1)}{2}}~,}</annotation></semantics></math></span><img alt="{\displaystyle 1+2+\cdots +n^{2}={\frac {n^{2}(n^{2}+1)}{2}}~,}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ee67e0bb074c10ad471c605e5830978184e6fc2c" style="border: 0px; display: inline-block; height: 5.843ex; vertical-align: -1.838ex; width: 32.069ex;" /></span></dd></dl>
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tandis que la <i>valeur</i> de ce carré, c'est-à-dire le même nombre que l'on retrouve en sommant les lignes, les colonnes, ou les deux diagonales vaut, puisqu'il y a <i>n</i> lignes et <i>n</i>colonnes, la quantité précédente divisée par <i>n</i> c'est-à-dire :</div>
<dl style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle {\frac {1+2+\cdots +n^{2}}{n}}={\frac {n(n^{2}+1)}{2}}~.}</annotation></semantics></math></span><img alt="{\displaystyle {\frac {1+2+\cdots +n^{2}}{n}}={\frac {n(n^{2}+1)}{2}}~.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f5fe3f6c54d025aac1ff9b690035e0c308ab2c1f" style="border: 0px; display: inline-block; height: 5.843ex; vertical-align: -1.838ex; width: 31.844ex;" /></span></dd></dl>
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Les différentes tailles n sont mises en correspondance avec les « cieux » dans les représentations traditionnelles. Le carré d'ordre 4, tel celui que l'on trouve dans la Melencholia, est associé au ciel de Jupiter. La somme de tous ses nombres vaut donc 136, et sa valeur est 34. Le carré d'ordre 3 correspond au ciel de Saturne. Le carré d'ordre 6 est traditionnellement associé au ciel du Soleil. La somme de tous ses nombres vaut donc <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><annotation encoding="application/x-tex">{\displaystyle 1+2+\cdots +6^{2}=1+2+\cdots +36=666}</annotation></semantics></math></span><img alt="{\displaystyle 1+2+\cdots +6^{2}=1+2+\cdots +36=666}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/05ef55f834649ce08ab9597eff69ee529b4c5825" style="border: 0px; display: inline-block; height: 2.843ex; margin: 0px; vertical-align: -0.505ex; width: 41.581ex;" /></span>, et sa valeur est 111. Ainsi, on retrouve le fait que 666 est avant tout considéré, notamment par la <a href="https://fr.wikipedia.org/wiki/Kabbale" style="background: none; color: #0b0080; text-decoration-line: none;" title="Kabbale">Kabbale</a>, comme un nombre « solaire », et c'est uniquement l'un de ses aspects, négatif, qui doit être considéré comme « maléfique », et non le nombre en lui-même, qui garde avant tout cet aspect solaire.</div>
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Le carré figurant dans la Melencholia est un type particulier de carré magique: la somme dans l'un de ses quatre quadrants, ainsi que la somme des nombres du carré du milieu, valent également 34, la valeur du carré<span class="reference" id="cite_ref-10" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Melencolia_de_D%C3%BCrer#cite_note-10" style="background: none; color: #0b0080; text-decoration-line: none;">10</a></span>. C'est un <b>carré magique gnomon</b>.</div><div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><br /></div><div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><br /></div><div style="background-color: white; color: #222222; font-family: sans-serif; font-size: 14px; line-height: inherit; margin-bottom: 0.5em; margin-top: 0.5em;"><h2 id="ref_2" style="border-bottom: 1px dotted rgb(0, 102, 221); color: #0066dd; font-family: Arial, Helvetica, sans-serif; font-size: 18px; margin-bottom: 10px; margin-top: 20px; max-width: 100%; text-align: justify;">Opérations</h2><p style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: justify;">La somme de deux carrés magiques du même ordre donne également un carré magique, mais le résultat n'est pas normal, c'est-à-dire que les nombres ne forment pas la suite 1, 2, 3... Également, deux carrés magiques du même ordre peuvent être soustraits.</p><p style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: justify;">Le « produit » de deux carrés magiques crée un carré magique d'ordre supérieur aux deux multiplicandes. Ce produit s'effectue ainsi. Soit les carrés magiques M et N :</p><ol style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: justify;"><li style="margin-bottom: 0.5em; max-width: 100%;">Le carré final sera d'ordre MxN.</li><li style="margin-bottom: 0.5em; max-width: 100%;">Diviser le damier final en NxN sous-damiers de MxM cases.</li><li style="margin-bottom: 0.5em; max-width: 100%;">Dans le carré N, réduire de 1 la valeur de tous les nombres.</li><li style="margin-bottom: 0.5em; max-width: 100%;">Multiplier ces valeurs réduites par M × M. Les résultats sont reportés dans les cases de chaque sous-damier correspondant du carré final.</li><li style="margin-bottom: 0.5em; max-width: 100%;">Les cases du carré M sont additionnées NxN fois aux cases du damier final.</li></ol><div class="center" style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: center; width: 870px;"><div class="thumb tnone" style="margin-left: auto; margin-right: auto; max-width: 100%;"><div class="thumbinner" style="background-color: #f1f1f1; line-height: 0px; margin-left: auto; margin-right: auto; max-width: 100%; width: 352px;"><img alt="" class="thumbimage" height="142" src="https://www.techno-science.net/illustration/Definition/350px/Magic-Squares---Multiplication---1.png" style="border: 0px; height: auto; margin-left: auto; margin-right: auto; max-width: 100%;" width="350" /><div class="thumbcaption" style="font-size: 15px; line-height: 19px; margin-left: auto; margin-right: auto; max-width: 100%; padding: 5px 10px; text-align: justify; width: fit-content;">Soit à effectuer le « produit » de ces deux carrés magiques, un de 3x3 et l'autre de 4x4. Le carré magique final sera de 12x12.</div></div></div></div><div class="center" style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: center; width: 870px;"><div class="thumb tnone" style="margin-left: auto; margin-right: auto; max-width: 100%;"><div class="thumbinner" style="background-color: #f1f1f1; line-height: 0px; margin-left: auto; margin-right: auto; max-width: 100%; width: 402px;"><img alt="" class="thumbimage" height="343" src="https://www.techno-science.net/illustration/Definition/400px/Magic-Squares---Multiplication---2.png" style="border: 0px; height: auto; margin-left: auto; margin-right: auto; max-width: 100%;" width="400" /><div class="thumbcaption" style="font-size: 15px; line-height: 19px; margin-left: auto; margin-right: auto; max-width: 100%; padding: 5px 10px; text-align: justify; width: fit-content;">Le carré magique de 3x3 est remplacé par le produit (3 × 3) et chaque <span class="lienGlossaire" style="margin-left: auto; margin-right: auto; max-width: 100%;"><a href="https://www.techno-science.net/glossaire-definition/Nombre.html" style="color: #0066dd; margin-left: auto; margin-right: auto; max-width: 100%; text-decoration-line: none;">nombre</a></span> dans le carré 4x4 est diminué de 1. Le damier final, de taille 12x12, est divisé en 4x4 sous-damiers, chacun ayant 3x3 cases. Chacune des cases s'obtient en multipliant (3 × 3) par l'une des cases du carré magique 4x4. Par exemple, 117 est le produit de 3 × 3 × 13. Ce carré est magique, mais n'est pas normal. La prochaine étape va « corriger » cette « anomalie ».</div></div></div></div><div class="center" style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: center; width: 870px;"><div class="thumb tnone" style="margin-left: auto; margin-right: auto; max-width: 100%;"><div class="thumbinner" style="background-color: #f1f1f1; line-height: 0px; margin-left: auto; margin-right: auto; max-width: 100%; width: 402px;"><img alt="" class="thumbimage" height="400" src="https://www.techno-science.net/illustration/Definition/400px/Magic-Squares---Multiplication---3.png" style="border: 0px; height: auto; margin-left: auto; margin-right: auto; max-width: 100%;" width="400" /><div class="thumbcaption" style="font-size: 15px; line-height: 19px; margin-left: auto; margin-right: auto; max-width: 100%; padding: 5px 10px; text-align: justify; width: fit-content;">Après 4x4 additions du carré 3x3, le carré final est magique et normal.</div></div></div></div><p style="font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: justify;">La <span class="lienGlossaire" style="max-width: 100%;"><a href="https://www.techno-science.net/glossaire-definition/Multiplication.html" style="color: #0066dd; max-width: 100%; text-decoration-line: none;">multiplication</a></span> de carrés magiques permet de générer des carrés magiques de plus grandes tailles. Cette technique produit plus rapidement des carrés de grande taille que la construction à l'aide de l'une des méthodes directes (celles de la Loubère ou de Strachey, par exemple).</p></div>
alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-37167603752404294942024-02-24T19:28:00.002+01:002024-02-24T19:28:12.010+01:00Nombres magiques<p> <span style="background-color: white; color: #333333; font-family: Roboto, sans-serif; font-size: 18px;">Les « nombres magiques » de protons et de neutrons peuvent rendre un noyau atomique particulièrement stable. </span></p><p><span style="background-color: white; color: #333333; font-family: Roboto, sans-serif; font-size: 18px;"><br /></span></p><p><span style="background-color: white; color: #333333; font-family: Roboto, sans-serif; font-size: 18px;">Les nombres magiques traditionnels sont 8, 20, 28, 50, 82 et 126.</span></p><p><span style="background-color: white; color: #333333; font-family: Roboto, sans-serif; font-size: 18px;"><br /></span></p><p><span style="background-color: white; color: #333333; font-family: Roboto, sans-serif; font-size: 18px;"> Dans des études antérieures, les chercheurs ont découvert la disparition des nombres magiques traditionnels et l’émergence de nouveaux nombres magiques du côté riche en neutrons de la carte des nucléides.</span></p><div id="974871025" style="background: 0px 0px rgb(255, 255, 255); border-style: solid; border-width: 0px; color: #333333; font-family: Roboto, sans-serif; font-size: 18px; margin: 0px auto; max-width: 662px; outline: 0px; padding: 0px; vertical-align: baseline;"></div><p style="background: 0px 0px rgb(255, 255, 255); border-style: solid; border-width: 0px; color: #333333; font-family: Roboto, sans-serif; font-size: 18px; margin: 0px auto; max-width: 662px; outline: 0px; padding: 0px; vertical-align: baseline;">D’autres nombres magiques traditionnels disparaîtront-ils dans les régions nucléaires extrêmement déficientes en neutrons ? </p><p style="background: 0px 0px rgb(255, 255, 255); border-style: solid; border-width: 0px; color: #333333; font-family: Roboto, sans-serif; font-size: 18px; margin: 0px auto; max-width: 662px; outline: 0px; padding: 0px; vertical-align: baseline;"><br /></p><p style="background: 0px 0px rgb(255, 255, 255); border-style: solid; border-width: 0px; color: #333333; font-family: Roboto, sans-serif; font-size: 18px; margin: 0px auto; max-width: 662px; outline: 0px; padding: 0px; vertical-align: baseline;">Une exploration plus approfondie est d’une grande importance pour enrichir et développer les théories nucléaires et approfondir notre compréhension des forces nucléaires.</p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-83007961445674927972024-02-23T07:38:00.005+01:002024-02-23T07:38:36.156+01:00LE QUANTIQUE, LES MATHEMATIQUES ET LE TEMPS<p></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="399" src="https://www.youtube.com/embed/tLdQqsWPAKI" width="480" youtube-src-id="tLdQqsWPAKI"></iframe></div><br /> <p></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-19207868713267440772024-02-03T17:56:00.000+01:002016-02-03T18:00:31.683+01:00COMBIEN Y A-T-IL D'ATOMES DANS L'UNIVERS ?<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-i_Nn8WfTf7Y/VrIxAwut8oI/AAAAAAABD-Y/HcAfRrcfPBY/s1600/univers.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="372" src="http://3.bp.blogspot.com/-i_Nn8WfTf7Y/VrIxAwut8oI/AAAAAAABD-Y/HcAfRrcfPBY/s640/univers.jpg" width="640" /></a></div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-ssPHiggUcvE/VrIxIHh9YpI/AAAAAAABD-c/26xFQlznx18/s1600/univers1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="366" src="http://4.bp.blogspot.com/-ssPHiggUcvE/VrIxIHh9YpI/AAAAAAABD-c/26xFQlznx18/s640/univers1.jpg" width="640" /></a></div>
<a href="http://www.lacosmo.com/dixpuissance80.html" target="_blank">Source</a><br />
Un divisé par 81 en base dix.<br />
Voir 1/<b><span style="font-size: x-large;">81</span></b>=0.0<b><span style="font-size: x-large;">123456790</span></b>123...... Dans ce blog .....alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-58636424278033303462023-11-14T08:14:00.005+01:002023-11-21T07:27:11.065+01:00Base de numération factoriele et décimale<p> 10!-1(base 10)=987654321(base factorielle)</p><p>3628800 (base 10)= 10^9 (base factorielle)</p><p>987654322 = 3628800 ou 1087654321 ou 9887654321 oi 9877654321 ou 987664321 ou 9876554321 ou 9876544321 ou 9876543321 (ou 9876543211 ou 1987654321 ou 19876543211)</p><p><br /></p><p>Le nombre exprime du temps.</p><p><br /></p><p>10! secondes = 6 semaines ..... Role du 7 dans 22/7==PI</p><p>et</p><p>22/<b>7</b>+333/<b>106</b>=355/<b>113 faux avec nos règles arithmétiques</b></p><p><b>22/7+ 333/106 =( 22+333)/(7+106)</b></p><p><b><br /></b></p><p><b><br /></b></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-39818337673986934532023-07-21T07:46:00.001+02:002023-07-21T07:46:19.236+02:00Congres Solvay<p></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7YmUFNoWEB7bz7lrsnALLhLyoZheyLESgPSlLlqk4YltRvRknDnTLA1XfZcRFjDICOBc2VJlBtcM_Z-77fTP7LnfJQGsMnEPdt6RzhYwDKKca_JzIR3Cy-u1qehIw1ibxufJ-tv3GqoW5X5uQi82joNVAXa9V949KZ2ItTfAMl_x_PFGAewA6a7Hdsw8/s1200/1998-02-berry2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="634" data-original-width="1200" height="306" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEi7YmUFNoWEB7bz7lrsnALLhLyoZheyLESgPSlLlqk4YltRvRknDnTLA1XfZcRFjDICOBc2VJlBtcM_Z-77fTP7LnfJQGsMnEPdt6RzhYwDKKca_JzIR3Cy-u1qehIw1ibxufJ-tv3GqoW5X5uQi82joNVAXa9V949KZ2ItTfAMl_x_PFGAewA6a7Hdsw8/w579-h306/1998-02-berry2.jpg" width="579" /></a></div><br /> <span style="background-color: #4a4a4a; color: white; font-family: franklin-gothic-urw, Arial, "Helvetica Neue", Helvetica, sans-serif; font-size: 14px;">The 1927 Solvay Congress in Brussels was attended by most of the leading physicists of the time. </span><p></p><p><span style="background-color: #4a4a4a; color: white; font-family: franklin-gothic-urw, Arial, "Helvetica Neue", Helvetica, sans-serif; font-size: 14px;">Dirac is in the second row, on Einstein's right. </span></p><p><span style="background-color: #4a4a4a; color: white; font-family: franklin-gothic-urw, Arial, "Helvetica Neue", Helvetica, sans-serif; font-size: 14px;">The other delegates are (left to right): front row</span></p><p><span style="background-color: #4a4a4a; color: white; font-family: franklin-gothic-urw, Arial, "Helvetica Neue", Helvetica, sans-serif; font-size: 14px;"> I Langmuir, M Planck, Madame Curie, H A Lorentz, A Einstein, P Langevin, Ch E Guye, CT R Wilson, 0 W Richardson;</span></p><p><span style="background-color: #4a4a4a; color: white; font-family: franklin-gothic-urw, Arial, "Helvetica Neue", Helvetica, sans-serif; font-size: 14px;"> second row: P Debye, M Knudsen,.W L Bragg, H A Kramers, P A M Dirac, A H Compton, L V de Broglie, M Born, N Bohr; back row; A Piccard, E Henriot, P Ehrenfest, E D Herzen, T H de Donder, E Schrodinger, E Verschaffelt, W Pauli, W Heisenberg, R H Fowler, L Brillouin.</span></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-45213341561697517122023-05-12T16:55:00.005+02:002023-05-12T16:55:37.856+02:00Vide<p> <span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">La</span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><b style="color: #202122; font-family: sans-serif; font-size: 14px;">perméabilité du vide</b><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">, également nommée</span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><b style="color: #202122; font-family: sans-serif; font-size: 14px;">perméabilité magnétique du vide</b><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">ou</span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><b style="color: #202122; font-family: sans-serif; font-size: 14px;">constante magnétique</b><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">, est une</span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><a href="https://fr.wikipedia.org/wiki/Constante_physique" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #3366cc; font-family: sans-serif; font-size: 14px; overflow-wrap: break-word; text-decoration-line: none;" title="Constante physique">constante physique</a><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">symbolisée par</span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><b style="color: #202122; font-family: sans-serif; font-size: 14px;">μ<sub style="line-height: 1;">0</sub></b><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">.</span></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Dans le système <a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_international_d%27unit%C3%A9s" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Système international d'unités">SI</a>, sa valeur est exactement :</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">μ<sub style="line-height: 1;">0</sub> = 4π × 10<sup style="line-height: 1;">−7</sup> <abbr class="abbr" style="border-bottom: 0px; cursor: help; text-decoration-line: none; text-decoration-style: initial;" title="kilogramme mètre par ampère carré seconde carré"><a href="https://fr.wikipedia.org/wiki/Kilogramme" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Kilogramme">kg</a> <a href="https://fr.wikipedia.org/wiki/M%C3%A8tre" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Mètre">m</a> <a href="https://fr.wikipedia.org/wiki/Amp%C3%A8re" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Ampère">A</a><sup style="line-height: 1;">−2</sup> <a href="https://fr.wikipedia.org/wiki/Seconde_(temps)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Seconde (temps)">s</a><sup style="line-height: 1;">−2</sup></abbr>, ou encore 4π × 10<sup style="line-height: 1;">−7</sup> <a href="https://fr.wikipedia.org/wiki/Tesla_(unit%C3%A9)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Tesla (unité)">T</a> m/A</dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">T étant le <a href="https://fr.wikipedia.org/wiki/Tesla_(unit%C3%A9)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Tesla (unité)">tesla</a>, unité d'induction électromagnétique</dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">soit donc :</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">μ<sub style="line-height: 1;">0</sub> = 12,566 370 614... × 10<sup style="line-height: 1;">−7</sup> <abbr class="abbr" style="border-bottom: 0px; cursor: help; text-decoration-line: none; text-decoration-style: initial;" title="kilogramme mètre par ampère carré seconde carré">kg m A<sup style="line-height: 1;">−2</sup> s<sup style="line-height: 1;">−2</sup></abbr>, ou encore 12,566 370 614... × 10<sup style="line-height: 1;">−7</sup> T m/A</dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">La constante magnétique est souvent exprimée en <a href="https://fr.wikipedia.org/wiki/Henry_(unit%C3%A9)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Henry (unité)">henry</a> par mètre : μ<sub style="line-height: 1;">0</sub> = 4π × 10<sup style="line-height: 1;">−7</sup> <abbr class="abbr" style="border-bottom: 0px; cursor: help; text-decoration-line: none; text-decoration-style: initial;" title="H par mètre">H m<sup style="line-height: 1;">−1</sup></abbr>.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">La valeur donnée était exacte par définition de l'<a href="https://fr.wikipedia.org/wiki/Amp%C3%A8re" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Ampère">ampère</a>, mais ne l'est plus depuis la redéfinition des <a href="https://fr.wikipedia.org/wiki/Unit%C3%A9s_de_base_du_Syst%C3%A8me_international" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Unités de base du Système international">unités</a> du <a href="https://fr.wikipedia.org/wiki/Syst%C3%A8me_international_d%27unit%C3%A9s" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Système international d'unités">système international</a>, le 20 mai 2019, la définition de l'ampère étant dorénavant liée à la définition de la <a href="https://fr.wikipedia.org/wiki/Charge_%C3%A9l%C3%A9mentaire" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Charge élémentaire">charge élémentaire</a> e qui a été choisie comme exacte, alors que la définition antérieure approuvée au Congrès Général des Poids et Mesures de 1948 fixait la perméabilité du vide<span class="reference" id="cite_ref-1" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Perm%C3%A9abilit%C3%A9_du_vide#cite_note-1" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;">1</a></span>.</p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-62451038166746629692023-05-12T16:53:00.000+02:002023-05-12T16:53:40.790+02:006 et 7<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9HWSA6eNGchfa9mQqmkxG-CBJxochwbtW8RcCq-a1KMguyGKsRP6HPVRKnpj_nP7IuuIIHI3utnjSI60kaSSFrZ--R1T81Pf5Oz0lY3aWpi-XfonSQGcpYxB56vzt2fa37-Fa2rlweFrQWAUpJanvj6V0Nu3VTY5o8CzUD71AQ2zkv3tiOTwOu-Jx/s1015/chiralite.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="429" data-original-width="1015" height="284" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg9HWSA6eNGchfa9mQqmkxG-CBJxochwbtW8RcCq-a1KMguyGKsRP6HPVRKnpj_nP7IuuIIHI3utnjSI60kaSSFrZ--R1T81Pf5Oz0lY3aWpi-XfonSQGcpYxB56vzt2fa37-Fa2rlweFrQWAUpJanvj6V0Nu3VTY5o8CzUD71AQ2zkv3tiOTwOu-Jx/w673-h284/chiralite.jpg" width="673" /></a></div><p></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">La <b>chiralité</b> (du <a href="https://fr.wikipedia.org/wiki/Grec_ancien" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Grec ancien">grec</a> χείρ, <i>kheir</i> : main) est une importante propriété reliant les notions de <a href="https://fr.wikipedia.org/wiki/Sym%C3%A9trie" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Symétrie">symétrie</a> et d'<a href="https://fr.wikipedia.org/wiki/Orientation_(math%C3%A9matiques)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Orientation (mathématiques)">orientation</a>, intervenant dans diverses branches de la science.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Un objet ou un système est appelé <b>chiral</b> s’il n'est pas superposable à son image dans un <a href="https://fr.wikipedia.org/wiki/Miroir" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Miroir">miroir</a>. Cet objet et son image miroir constituent alors deux formes différentes qualifiées d'<b>énantiomorphes</b> (du grec <i>formes opposées</i>) ou, en se référant à des <a href="https://fr.wikipedia.org/wiki/Mol%C3%A9cule" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Molécule">molécules</a>, des conformations spatiales « <a href="https://fr.wikipedia.org/wiki/L%C3%A9vogyre" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Lévogyre">gauches</a> » et « <a href="https://fr.wikipedia.org/wiki/Dextrogyre" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Dextrogyre">droites</a> » appelées <a href="https://fr.wikipedia.org/wiki/%C3%89nantiom%C3%A9rie" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Énantiomérie">énantiomères</a> dotés d'une asymétrie moléculaire tridimensionnelle. Le groupe des isométries laissant globalement invariant l'objet initial ne possède que des <a href="https://fr.wikipedia.org/wiki/Rotation_affine" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Rotation affine">rotations</a>.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Un objet (ou une molécule) non chiral est dit <b>achiral</b> (ou parfois <b>amphichiral</b>). Il est superposable à son image miroir. Autrement dit, le groupe des isométries laissant globalement <a href="https://fr.wikipedia.org/wiki/Invariant" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Invariant">invariant</a> l'objet possède au moins une isométrie indirecte.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">La <b>chirogenèse</b> est la formation de molécules chirales.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd4kRZf6jY88sEjntOyQ2ISsNx9QiOTIt1YbF8gMD7qmYjCKQLND4EIujPDMoDdQLeOgt9byN6yK3KO_f2K7mCi8coc6rqfHWbOLI1ZbIJC0wXJiGk-ItKAJmjq6AJYzPq2n8BRy_3PB0o2AZZhTqsp_KmCUb78Gr4c-JGlBO1A-QaTIfQO3A7CrDs/s712/chiralite1.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="712" data-original-width="555" height="583" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhd4kRZf6jY88sEjntOyQ2ISsNx9QiOTIt1YbF8gMD7qmYjCKQLND4EIujPDMoDdQLeOgt9byN6yK3KO_f2K7mCi8coc6rqfHWbOLI1ZbIJC0wXJiGk-ItKAJmjq6AJYzPq2n8BRy_3PB0o2AZZhTqsp_KmCUb78Gr4c-JGlBO1A-QaTIfQO3A7CrDs/w454-h583/chiralite1.jpg" width="454" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixv3vvLLcTbFMb27E25RavAYBvvzjCOn5lDM9ZOX83eSgORS0jCQoVTNqGvI4WcABDT6TXtrYA_yy884qyl19ddM-P7t6BbroGM7nWXu1xAHmJayxBsS5zJY_7CiVQTRMSXNg38DlvITT54czPOSHWd5PIQ4PY-_SryjmvhmMRkJBxFNnPmOjxWyAm/s812/chiralite2.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="479" data-original-width="812" height="309" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEixv3vvLLcTbFMb27E25RavAYBvvzjCOn5lDM9ZOX83eSgORS0jCQoVTNqGvI4WcABDT6TXtrYA_yy884qyl19ddM-P7t6BbroGM7nWXu1xAHmJayxBsS5zJY_7CiVQTRMSXNg38DlvITT54czPOSHWd5PIQ4PY-_SryjmvhmMRkJBxFNnPmOjxWyAm/w524-h309/chiralite2.jpg" width="524" /></a></div><br /><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-33009566539943839522023-05-12T08:20:00.000+02:002023-05-12T08:20:00.997+02:00Classification de Chancourtois<p> </p><h3 style="background-color: white; box-sizing: border-box; color: #003e5c; font-family: Roboto; font-size: 18px; font-weight: normal; line-height: 1.1; margin-bottom: 9.5px; margin-top: 19px;">La vis tellurique</h3><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;">Chancourtois a nommé sa classification vis tellurique : « D’après son mode de réalisation et son origine, je lui donne le nom significatif de vis tellurique », écrit-il dans le rapport à l’Académie des sciences du 7 avril 1862. Un peu plus tard, le 5 mai de la même année, tout en précisant que le nom lui a été suggéré surtout par la place centrale de l’élément <i style="box-sizing: border-box;">tellure</i> sur la vis, il écrit que « l’épithète tellurique (…) rappelle très heureusement l’origine géognostique, puisque <i style="box-sizing: border-box;">tellus</i> signifie terre dans le sens le plus positif, le plus familier, dans le sens de terre nourricière ». Ce propos montre bien que la géologie était le point de départ des réflexions de Chancourtois. Il voyait la forme hélicoïdale comme étant idéale pour représenter une périodicité : ainsi, il a utilisé des méthodes comparables pour rechercher une logique mathématique dans les relations entre les différentes formations géologiques de la Terre.</p><h3 style="background-color: white; box-sizing: border-box; color: #003e5c; font-family: Roboto; font-size: 18px; font-weight: normal; line-height: 1.1; margin-bottom: 9.5px; margin-top: 19px;">Priorité aux nombres</h3><p class="p2" style="background-color: white; box-sizing: border-box; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><span style="color: red;">Chancourtois avait aussi une vision mathématique de la matière,</span></p><p class="p2" style="background-color: white; box-sizing: border-box; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><span style="color: red;"> selon laquelle « les propriétés des corps sont les propriétés des nombres » (cf. son ouvrage de 1863, <i style="box-sizing: border-box;">La Vis tellurique</i>). </span></p><p class="p2" style="background-color: white; box-sizing: border-box; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><span style="color: red;">Cette vision pourrait expliquer la priorité qu’il donnait aux nombres plutôt qu’aux appartenances à une famille chimique.</span></p><p class="p2" style="background-color: white; box-sizing: border-box; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><span style="color: red;"></span><span style="color: #212121;">Ainsi Chancourtois n’a pas correctement placé l’<i style="box-sizing: border-box;">iode</i> en dessous du <i style="box-sizing: border-box;">chlore</i> et du <i style="box-sizing: border-box;">fluor</i> comme l’ont fait ses successeurs : le piège, c’est que, curieusement, l’<i style="box-sizing: border-box;">iode </i>(aujourd’hui élément 53) est plus léger que le <i style="box-sizing: border-box;">tellure</i> (aujourd’hui élément 52). De plus, cette vision mathématique le conduisait à tenter de prédire des propriétés chimiques à partir d’une factorisation des masses des éléments, supposées entières. Dans cet esprit, il aurait aimé rapprocher la notion d’élément de celle de nombre premier.</span></p><h3 style="background-color: white; box-sizing: border-box; color: #003e5c; font-family: Roboto; font-size: 18px; font-weight: normal; line-height: 1.1; margin-bottom: 9.5px; margin-top: 19px;">Une classification imparfaite</h3><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><span class="s1" style="box-sizing: border-box;">L’approche étonnante de Chancourtois et les nombreuses imperfections dans sa classification pourraient expliquer en partie le fait que la vis tellurique n’ait pas eu autant de reconnaissance que le tableau de Mendeleïev. Pourtant, cette première classification périodique a ses mérites. L’importance du numéro atomique aujourd’hui confirme en quelque sorte l’intuition de Chancourtois qui voyait un rapport étroit entre les nombres et la nature des corps. De même, son idée que ces nombres pouvaient servir à prédire et à expliquer les spectres de raies des éléments avait un côté prophétique : c’est au fond ce qu’apportera en 1913 la loi de Moseley reliant les fréquences (</span><span class="s2" style="box-sizing: border-box;">ν</span><span class="s1" style="box-sizing: border-box;">) des raies au numéro atomique (Z) de l’élément. Toutefois, l’intérêt de cette vis tellurique n’a pas été véritablement perçu à l’époque, d’autant plus que Chancourtois ne faisait pas partie du cercle des chimistes.</span></p><p style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"></p><hr style="background-color: white; border-bottom: 0px rgb(229, 229, 229); border-image: initial; border-left: 0px rgb(229, 229, 229); border-right: 0px rgb(229, 229, 229); border-top-color: rgb(229, 229, 229); border-top-style: solid; box-sizing: content-box; color: #003e5c; font-family: Roboto; font-size: 15px; height: 0px; margin-bottom: 19px; margin-top: 19px;" /><h3 class="p1" style="background-color: white; box-sizing: border-box; color: #003e5c; font-family: Roboto; font-size: 18px; font-weight: normal; line-height: 1.1; margin-bottom: 9.5px; margin-top: 19px;"><span class="s1" style="box-sizing: border-box;">Les triades entrent en scène</span></h3><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;">En 1817, le chimiste allemand Döbereiner identifie une première « triade » : </p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;">trois éléments alcalinoterreux (calcium, strontium et baryum)</p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><br /></p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"> dont la masse de l’élément du milieu est égale à la moyenne des masses des deux autres.</p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><br /></p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"> Ce concept prend corps avec trois autres triades publiées en 1829, alignant des éléments qui sont superposés dans le tableau périodique actuel. C’est la première découverte de rapports quantitatifs entre les masses d’éléments d’une même famille, et donc en quelque sorte un premier pas vers le tableau périodique actuel, où ces rapports se comprennent directement. En 1843, Gmelin combine pour la première fois des triades (c’est d’ailleurs lui qui a trouvé ce nom) dans un tableau comportant 55 éléments. Bien qu’on ne puisse pas encore considérer cette classification comme étant périodique, elle regroupait déjà correctement les éléments des futures colonnes 1, 2, 15, 16 et 17 du tableau actuel, du moins pour ses trois premières lignes.</p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><br /></p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><a href="https://www.lajauneetlarouge.com/vis-tellurique-chancourtois/" target="_blank">Source</a><br /></p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><br /></p><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWtErX-AqiDRpdyK4Bv3z7s9X5HHIzyKk_sHPVO2t-cHsBNU9naDaKbbIDfQ3abX20ppfeuG_xtAKrpOdcwcLIGtjFIuYatpLRkY8qzHxX5EFiVOaDjafCw_PdrVpRMfBjj8X8jwxJp9P0GOddiXUQxax7hvcgyvAGV-1Xvb1jHWIrH7F8xbgWYcFC/s1118/Hijmans-Avenas_figure_5.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="1118" data-original-width="850" height="575" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgWtErX-AqiDRpdyK4Bv3z7s9X5HHIzyKk_sHPVO2t-cHsBNU9naDaKbbIDfQ3abX20ppfeuG_xtAKrpOdcwcLIGtjFIuYatpLRkY8qzHxX5EFiVOaDjafCw_PdrVpRMfBjj8X8jwxJp9P0GOddiXUQxax7hvcgyvAGV-1Xvb1jHWIrH7F8xbgWYcFC/w437-h575/Hijmans-Avenas_figure_5.jpg" width="437" /></a></div><br /><p class="p2" style="background-color: white; box-sizing: border-box; color: #212121; font-family: Roboto; font-size: 15px; margin: 0px 0px 9.5px; text-rendering: optimizelegibility;"><br /></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-8620552767320542412023-05-12T07:31:00.002+02:002023-05-12T07:31:14.892+02:00temps et Euler<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhgaZc0S0icITeAgOZUkP5PG5yfSqM_R4cD9WWSSFvDHVEVLCMMvdeVUN242HmwtQQM5ZqtDbp4epxa6ocVxklcT3w7RMsqgLqMuqwrcH5ipQwOr1ZP__knjJiV0WZjz5x6LGKTxWilvS5RsqtefXEibQJT5Q51lD38IX3GByDXagdhEYFjplHjQHO/s998/euler.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="399" data-original-width="998" height="258" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhhgaZc0S0icITeAgOZUkP5PG5yfSqM_R4cD9WWSSFvDHVEVLCMMvdeVUN242HmwtQQM5ZqtDbp4epxa6ocVxklcT3w7RMsqgLqMuqwrcH5ipQwOr1ZP__knjJiV0WZjz5x6LGKTxWilvS5RsqtefXEibQJT5Q51lD38IX3GByDXagdhEYFjplHjQHO/w646-h258/euler.jpg" width="646" /></a></div><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">Parmi les rationnels de numérateur et dénominateur inférieurs à 1 000, le plus proche de </span><span class="texhtml" style="background-color: white; color: #202122; font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">e</span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> est</span><span class="reference" id="cite_ref-26" style="background-color: white; color: #202122; font-family: sans-serif; font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/E_(nombre)#cite_note-26" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;">18</a></span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> </span><span class="texhtml" style="background-color: white; color: #202122; font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><span style="display: inline-block; font-size: 14.042px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">878</span><span style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">323</span></span></span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"> ≈ </span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; white-space: nowrap;">2,718<span style="margin-left: 0.25em;">27</span></span><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;">.</span><p></p><p><span style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"><br /></span></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">En <a href="https://fr.wikipedia.org/wiki/Math%C3%A9matiques" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Mathématiques">mathématiques</a>, l'<b>identité d'Euler</b> est une relation entre plusieurs <a href="https://fr.wikipedia.org/wiki/Table_de_constantes_math%C3%A9matiques" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Table de constantes mathématiques">constantes</a> fondamentales et utilisant les trois opérations arithmétiques d'<a href="https://fr.wikipedia.org/wiki/Addition" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Addition">addition</a>, <a href="https://fr.wikipedia.org/wiki/Multiplication" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Multiplication">multiplication</a> et <a href="https://fr.wikipedia.org/wiki/Exponentiation" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Exponentiation">exponentiation</a> :</p><center style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \mathrm {e} ^{\mathrm {i} \pi }+1=0}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">e</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">i</mi></mrow><mi>�</mi></mrow></msup><mo>+</mo><mn>1</mn><mo>=</mo><mn>0</mn></mstyle></mrow></semantics></math></span><img alt="{\displaystyle \mathrm {e} ^{\mathrm {i} \pi }+1=0}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2bca7543787faa55f2781b4d3df77aaed2a7fdb" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.505ex; width: 10.928ex;" /></span></center><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">où la <a href="https://fr.wikipedia.org/wiki/E_(nombre)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="E (nombre)">base <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">e</span> du logarithme naturel</a> représente l'<a href="https://fr.wikipedia.org/wiki/Analyse_(math%C3%A9matiques)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Analyse (mathématiques)">analyse</a>, l'<a href="https://fr.wikipedia.org/wiki/Unit%C3%A9_imaginaire" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Unité imaginaire">unité imaginaire</a> <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">i</span> représente l'<a href="https://fr.wikipedia.org/wiki/Alg%C3%A8bre" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Algèbre">algèbre</a>, la <a href="https://fr.wikipedia.org/wiki/Pi" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Pi">constante d'Archimède</a> <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">π</span> représente la <a href="https://fr.wikipedia.org/wiki/G%C3%A9om%C3%A9trie" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Géométrie">géométrie</a>, <span class="need_ref" style="border-bottom: 1px solid rgb(170, 170, 170); cursor: help;" title="Une source est souhaitée pour ce passage (demandé le janvier 2015).">l'entier <a href="https://fr.wikipedia.org/wiki/1_(nombre)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="1 (nombre)">1</a> l'<a href="https://fr.wikipedia.org/wiki/Arithm%C3%A9tique" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Arithmétique">arithmétique</a> et le <a href="https://fr.wikipedia.org/wiki/Z%C3%A9ro" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Zéro"><span class="nowrap" style="white-space: nowrap;">nombre 0</span></a> les mathématiques</span><sup class="need_ref_tag" style="line-height: 1; padding-left: 2px;"><a href="https://fr.wikipedia.org/wiki/Aide:R%C3%A9f%C3%A9rence_n%C3%A9cessaire" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Aide:Référence nécessaire">[<abbr class="abbr" style="border-bottom: 0px; cursor: help; text-decoration-line: none; text-decoration-style: initial;" title="référence">réf.</abbr> souhaitée]</a></sup><span class="reference" id="cite_ref-1" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Identit%C3%A9_d%27Euler#cite_note-1" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;">1</a></span>.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Elle est nommée d'après le <a href="https://fr.wikipedia.org/wiki/Math%C3%A9maticien" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Mathématicien">mathématicien</a> <a href="https://fr.wikipedia.org/wiki/Leonhard_Euler" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Leonhard Euler">Leonhard Euler</a> qui la fait apparaître dans son <i><a href="https://fr.wikipedia.org/wiki/Introductio_in_analysin_infinitorum" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Introductio in analysin infinitorum">Introductio</a></i>, publié à <a href="https://fr.wikipedia.org/wiki/Lausanne" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Lausanne">Lausanne</a> en <a href="https://fr.wikipedia.org/wiki/1748" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="1748">1748</a>. Avant d'être citée par Euler, cette formule était connue du mathématicien anglais <a href="https://fr.wikipedia.org/wiki/Roger_Cotes" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Roger Cotes">Roger Cotes</a>, mort en 1716.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">L'identité d'Euler est souvent citée comme un exemple de <a href="https://fr.wikipedia.org/wiki/Beaut%C3%A9_math%C3%A9matique" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Beauté mathématique">beauté mathématique</a><span class="reference" id="cite_ref-Gallagher2014_3-0" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Identit%C3%A9_d%27Euler#cite_note-Gallagher2014-3" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;">3</a></span>.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">En effet, outre l'égalité, trois des opérations fondamentales de l'arithmétique y sont utilisées, chacune une fois : l'<a href="https://fr.wikipedia.org/wiki/Addition" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Addition">addition</a>, la <a href="https://fr.wikipedia.org/wiki/Multiplication" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Multiplication">multiplication</a> et l'<a href="https://fr.wikipedia.org/wiki/Exponentiation" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Exponentiation">exponentiation</a>. L'identité fait également intervenir cinq <a class="mw-redirect" href="https://fr.wikipedia.org/wiki/Constante_math%C3%A9matique" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Constante mathématique">constantes mathématiques</a> fondamentales<span class="reference" id="cite_ref-4" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Identit%C3%A9_d%27Euler#cite_note-4" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;">4</a></span> :</p><ul style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; list-style-image: url("/w/skins/Vector/resources/common/images/bullet-icon.svg?d4515"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;"><a class="mw-redirect" href="https://fr.wikipedia.org/wiki/0_(nombre)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="0 (nombre)">0</a>, l'élément neutre de l'addition.</li><li style="margin-bottom: 0.1em;"><a href="https://fr.wikipedia.org/wiki/1_(nombre)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="1 (nombre)">1</a>, l'élément neutre de la multiplication.</li><li style="margin-bottom: 0.1em;"><a href="https://fr.wikipedia.org/wiki/Pi" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Pi"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">π</span></a>, omniprésente en <a href="https://fr.wikipedia.org/wiki/Trigonom%C3%A9trie" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Trigonométrie">trigonométrie</a>, en géométrie dans l'<a href="https://fr.wikipedia.org/wiki/Espace_euclidien" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Espace euclidien">espace euclidien</a> et en <a class="mw-redirect" href="https://fr.wikipedia.org/wiki/Analyse_math%C3%A9matique" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Analyse mathématique">analyse mathématique</a> (<span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">π</span> = 3,14159265...)</li><li style="margin-bottom: 0.1em;"><a href="https://fr.wikipedia.org/wiki/E_(nombre)" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="E (nombre)"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">e</span></a>, base des <a href="https://fr.wikipedia.org/wiki/Logarithme" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Logarithme">logarithmes</a> qui apparait souvent en analyse, calcul différentiel et mathématiques financières (<span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">e</span> = 2,718281828...). Tout comme <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">π</span>, c'est un <a href="https://fr.wikipedia.org/wiki/Nombre_transcendant" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Nombre transcendant">nombre transcendant</a>.</li><li style="margin-bottom: 0.1em;"><a href="https://fr.wikipedia.org/wiki/Unit%C3%A9_imaginaire" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Unité imaginaire"><span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">i</span></a>, l'unité imaginaire à la base des <a href="https://fr.wikipedia.org/wiki/Nombre_complexe" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Nombre complexe">nombres complexes</a>, qui ont permis l'étude de la résolution des équations polynomiales avant de voir leur usage élargi.</li></ul><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">L'inventaire de ces différents éléments est mieux mis en évidence par la <a href="https://fr.wikipedia.org/wiki/Notation_polonaise_inverse" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Notation polonaise inverse">notation polonaise inverse</a> de la formule d'Euler :</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">0 ; 1 ; <i>e</i> ; <i>i</i> ;π ; * ; ^ ; +; =</dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">De plus, sous cette forme, l'identité est écrite comme une expression égale à zéro, une pratique courante en mathématique.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">On en déduit que l'<a href="https://fr.wikipedia.org/wiki/Exponentielle_complexe" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Exponentielle complexe">exponentielle complexe</a> est <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">2πi</span>-<a href="https://fr.wikipedia.org/wiki/Fonction_p%C3%A9riodique" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Fonction périodique">périodique</a>.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">On voit <i>ainsi</i> apparaître les développements en série de Taylor des fonctions cosinus et sinus<span class="reference" id="cite_ref-5" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Formule_d%27Euler#cite_note-5" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;">5</a></span> :</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \cos(x)=1-{\frac {x^{2}}{2\,!}}+{\frac {x^{4}}{4\,!}}-{\frac {x^{6}}{6\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k}}{(2k)\,!}}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mrow><mn>2</mn><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mrow><mn>4</mn><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>6</mn></mrow></msup><mrow><mn>6</mn><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow><mo>+</mo><mo>⋯</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msup><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>�</mi></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>�</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow></mstyle></mrow></semantics></math></span><img alt="\cos(x)=1-{\frac {x^{2}}{2\,!}}+{\frac {x^{4}}{4\,!}}-{\frac {x^{6}}{6\,!}}+\cdots =\sum _{{k=0}}^{\infty }{\frac {(-1)^{k}x^{{2k}}}{(2k)\,!}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3c54a5e9909aafd34b3581cb6a5de2923634512e" style="border: 0px; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 51.042ex;" /></span></dd></dl><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \sin(x)=x-{\frac {x^{3}}{3\,!}}+{\frac {x^{5}}{5\,!}}-{\frac {x^{7}}{7\,!}}+\cdots =\sum _{k=0}^{\infty }{\frac {(-1)^{k}x^{2k+1}}{(2k+1)\,!}}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>=</mo><mi>�</mi><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup><mrow><mn>3</mn><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msup><mrow><mn>5</mn><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>7</mn></mrow></msup><mrow><mn>7</mn><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow><mo>+</mo><mo>⋯</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>=</mo><mn>0</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msup><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn><mi>�</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><mrow><mo stretchy="false">(</mo><mn>2</mn><mi>�</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>!</mo></mrow></mfrac></mrow></mstyle></mrow></semantics></math></span><img alt="\sin(x)=x-{\frac {x^{3}}{3\,!}}+{\frac {x^{5}}{5\,!}}-{\frac {x^{7}}{7\,!}}+\cdots =\sum _{{k=0}}^{\infty }{\frac {(-1)^{k}x^{{2k+1}}}{(2k+1)\,!}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7220a6a47777abc45945bbb301252757a2f45c31" style="border: 0px; display: inline-block; height: 7.176ex; vertical-align: -3.171ex; width: 53.054ex;" /></span></dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">ce qui, en remplaçant dans l'expression précédente de <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">e<sup style="line-height: 1;">i<i>x</i></sup></span>, donne bien :</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \mathrm {e} ^{\mathrm {i} \,x}=\cos(x)+\mathrm {i} \,\sin(x).}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msup><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">e</mi></mrow><mrow class="MJX-TeXAtom-ORD"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">i</mi></mrow><mspace width="thinmathspace"></mspace><mi>�</mi></mrow></msup><mo>=</mo><mi>cos</mi><mo></mo><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">i</mi></mrow><mspace width="thinmathspace"></mspace><mi>sin</mi><mo></mo><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>.</mo></mstyle></mrow></semantics></math></span><img alt="{\mathrm {e}}^{{{\mathrm {i}}\,x}}=\cos(x)+{\mathrm {i}}\,\sin(x)." aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/08363db1783f1dacbda80b05b41c5b006c34c2ca" style="border: 0px; display: inline-block; height: 3.176ex; vertical-align: -0.838ex; width: 23.301ex;" /></span></dd></dl>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-32788366740149565232023-05-12T07:19:00.001+02:002023-05-19T06:45:14.867+02:00Un et:ou zéro<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6dlQ3n8-mnQxPiQXtgWYICzMPSG7bcQz4MXxDYdt6ZD3-6Lp07WoIoIx0FZelD8Be2uyDzVGPhaqIjOzOhaCXQzvfkNogyRhAkzg0TX2KV3vmltBzxQ-mZmFihQxu8rzf5ca2sEdu5lAS5YydrIZYV-WwZSeLPo9CR0CUK7NF74Fazw17ygReZ82W/s4128/20230510_071602.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="3096" data-original-width="4128" height="351" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg6dlQ3n8-mnQxPiQXtgWYICzMPSG7bcQz4MXxDYdt6ZD3-6Lp07WoIoIx0FZelD8Be2uyDzVGPhaqIjOzOhaCXQzvfkNogyRhAkzg0TX2KV3vmltBzxQ-mZmFihQxu8rzf5ca2sEdu5lAS5YydrIZYV-WwZSeLPo9CR0CUK7NF74Fazw17ygReZ82W/w468-h351/20230510_071602.jpg" width="468" /></a></div><br /><br /><p></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-58046697099257952862023-05-09T15:09:00.012+02:002023-05-09T15:23:38.649+02:00Cardinal d'un ensemble<p> <span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">Par exemple, soit</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><i style="color: #202122; font-family: sans-serif; font-size: 14px;">f</i><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">(</span><i style="color: #202122; font-family: sans-serif; font-size: 14px;">n</i><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">) le nombre de sous-ensembles distincts de nombres entiers dans l'intervalle [1,</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><i style="color: #202122; font-family: sans-serif; font-size: 14px;">n</i><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">] qui ne contiennent pas deux nombres entiers consécutifs. </span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">Par exemple, avec</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><i style="color: #202122; font-family: sans-serif; font-size: 14px;">n</i><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">= 4, nous obtenons ∅, { 1 }, { 2 }, { 3 }, { 4 }, { 1, 3 }, { 1, 4 }, { 2, 4 }, et donc f(4) = 8.</span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> Il s'avère que</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><i style="color: #202122; font-family: sans-serif; font-size: 14px;">f</i><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">(</span><i style="color: #202122; font-family: sans-serif; font-size: 14px;">n</i><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">) est le</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><i style="color: #202122; font-family: sans-serif; font-size: 14px;">n</i><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">ème</span><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> </span><a class="mw-redirect" href="https://fr.wikipedia.org/wiki/Nombre_de_Fibonacci" style="background-attachment: initial; background-clip: initial; background-image: none; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; color: #3366cc; font-family: sans-serif; font-size: 14px; overflow-wrap: break-word; text-decoration-line: none;" title="Nombre de Fibonacci">nombre de Fibonacci</a><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">, qui peut être exprimé sous la forme « fermée » suivante :</span></p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle f(n)={\frac {\phi ^{n}}{\sqrt {5}}}-{\frac {(1-\phi )^{n}}{\sqrt {5}}}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msup><msqrt><mn>5</mn></msqrt></mfrac></mrow><mo>−</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>�</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msup></mrow><msqrt><mn>5</mn></msqrt></mfrac></mrow></mstyle></mrow></semantics></math></span><img alt="f(n)={\frac {\phi ^{n}}{{\sqrt {5}}}}-{\frac {(1-\phi )^{n}}{{\sqrt {5}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5cedf7cbaccb5a672e91e62e1ebb303b281a0f87" style="border: 0px; display: inline-block; height: 6.676ex; vertical-align: -2.838ex; width: 23.608ex;" /></span></dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">où <i>Φ</i> = (1 + √5)/2, est le nombre d'or. Cependant, étant donné que nous considérons des ensembles de nombres entiers, la présence du √5 dans le résultat peut être considérée comme inesthétique d'un point de vue combinatoire. Aussi <i>f</i>(<i>n</i>) peut-il être exprimé par une relation de récurrence :</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>f</i>(<i>n</i>) = <i>f</i>(<i>n</i> - 1) + <i>f</i> (<i>n</i> - 2)</dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">ce qui peut être plus satisfaisant (d'un point de vue purement combinatoire), puisque la relation montre plus clairement comment le résultat a été trouvé.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Dans certains cas, un équivalent asymptotique <i>g</i> de <i>f</i>,</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><i>f</i>(<i>n</i>)~<i>g</i>(<i>n</i>) quand <i>n</i> tend vers l'infini</dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">où <i>g</i> est une fonction « familière », permet d'obtenir une bonne approximation de <i>f</i>. Une fonction asymptotique simple peut être préférable à une formule « fermée » extrêmement compliquée et qui informe peu sur le comportement du nombre d'objets. Dans l'exemple ci-dessus, un équivalent asymptotique serait :</p><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle f(n)\sim {\frac {\phi ^{n}}{\sqrt {5}}}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>∼</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msup><msqrt><mn>5</mn></msqrt></mfrac></mrow></mstyle></mrow></semantics></math></span><img alt="f(n)\sim {\frac {\phi ^{n}}{{\sqrt {5}}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f8298da336a8b779b174b002687200b79d5291ef" style="border: 0px; display: inline-block; height: 6.343ex; vertical-align: -2.838ex; width: 11.516ex;" /></span></dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">quand <i>n</i> devient grand.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Une autre approche est celle des séries entières. <i>f</i>(<i>n</i>) peut être exprimé par une série entière formelle, appelée fonction génératrice de <i>f</i>, qui peut être le plus couramment :</p><ul style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; list-style-image: url("/w/skins/Vector/resources/common/images/bullet-icon.svg?d4515"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">la fonction génératrice ordinaire</li></ul><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \sum _{n>0}f(n)\cdot x^{n}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>></mo><mn>0</mn></mrow></munder><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>⋅</mo><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msup></mstyle></mrow></semantics></math></span><img alt="\sum _{{n>0}}f(n)\cdot x^{n}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8959d03ab1e4f172bc36f88fe949517c6c6db44d" style="border: 0px; display: inline-block; height: 5.509ex; vertical-align: -3.005ex; width: 12.452ex;" /></span></dd></dl><ul style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; list-style-image: url("/w/skins/Vector/resources/common/images/bullet-icon.svg?d4515"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">ou la fonction génératrice exponentielle</li></ul><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \sum _{n>0}f(n)\cdot {\frac {x^{n}}{n!}}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><munder><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>></mo><mn>0</mn></mrow></munder><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>⋅</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msup><mrow><mi>�</mi><mo>!</mo></mrow></mfrac></mrow></mstyle></mrow></semantics></math></span><img alt="\sum _{{n>0}}f(n)\cdot {\frac {x^{n}}{n!}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e3c78ec4cbf7b4a2d62ede7981d49907ee2aa584" style="border: 0px; display: inline-block; height: 6.343ex; vertical-align: -3.005ex; width: 13.288ex;" /></span></dd></dl><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Une fois déterminée, la fonction génératrice peut permettre d'obtenir toutes les informations fournies par les approches précédentes. En outre, les diverses opérations usuelles comme l'addition, la multiplication, la dérivation, etc., ont une signification combinatoire ; et ceci permet de prolonger des résultats d'un problème combinatoire afin de résoudre d'autres problèmes.</p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><a href="https://fr.wikipedia.org/wiki/Combinatoire">Combinatoire — Wikipédia (wikipedia.org)</a></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p><ul style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; list-style-image: url("/w/skins/Vector/resources/common/images/bullet-icon.svg?d4515"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">Un produit cartésien <i>A × B</i> est <a href="https://fr.wikipedia.org/wiki/Ensemble_vide" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Ensemble vide">vide</a> si et seulement si <i>A</i> ou <i>B</i> est vide. En particulier : pour tout ensemble <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle A}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi></mstyle></mrow></semantics></math></span><img alt="A" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 1.743ex;" /></span>,<center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \varnothing \times A=A\times \varnothing =\varnothing }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi class="MJX-variant">∅</mi><mo>×</mo><mi>�</mi><mo>=</mo><mi>�</mi><mo>×</mo><mi class="MJX-variant">∅</mi><mo>=</mo><mi class="MJX-variant">∅</mi></mstyle></mrow></semantics></math></span><img alt="{\displaystyle \varnothing \times A=A\times \varnothing =\varnothing }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a619bf6f18440bd9ac3ce9bf0e524b9ed3687c87" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 20.788ex;" /></span>.</center></li><li style="margin-bottom: 0.1em;">Les deux facteurs d'un produit sont entièrement déterminés par ce produit, s'il est non vide. Plus précisément : si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle A\neq \varnothing }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo>≠</mo><mi class="MJX-variant">∅</mi></mstyle></mrow></semantics></math></span><img alt="{\displaystyle A\neq \varnothing }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0c84bfaec7b2091189e67d3e979e4474a35640e" style="border: 0px; display: inline-block; height: 2.676ex; vertical-align: -0.838ex; width: 6.65ex;" /></span> alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle y\in B\Leftrightarrow \exists x\quad (x,y)\in A\times B}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo>∈</mo><mi>�</mi><mo stretchy="false">⇔</mo><mi mathvariant="normal">∃</mi><mi>�</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>�</mi><mo>,</mo><mi>�</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>�</mi><mo>×</mo><mi>�</mi></mstyle></mrow></semantics></math></span><img alt="{\displaystyle y\in B\Leftrightarrow \exists x\quad (x,y)\in A\times B}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a27420dd333e3f2d078bb4697fcae3ac6f1582a3" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 28.836ex;" /></span> et de même, si <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle B\neq \varnothing }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo>≠</mo><mi class="MJX-variant">∅</mi></mstyle></mrow></semantics></math></span><img alt="{\displaystyle B\neq \varnothing }" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a80fef2c17659c6e85c3401e327708f60398513" style="border: 0px; display: inline-block; height: 2.676ex; vertical-align: -0.838ex; width: 6.671ex;" /></span> alors <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle x\in A\Leftrightarrow \exists y\quad (x,y)\in A\times B}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo>∈</mo><mi>�</mi><mo stretchy="false">⇔</mo><mi mathvariant="normal">∃</mi><mi>�</mi><mspace width="1em"></mspace><mo stretchy="false">(</mo><mi>�</mi><mo>,</mo><mi>�</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>�</mi><mo>×</mo><mi>�</mi></mstyle></mrow></semantics></math></span><img alt="{\displaystyle x\in A\Leftrightarrow \exists y\quad (x,y)\in A\times B}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2531b0fe9dbcc192f24fef1455f88dc28aecb732" style="border: 0px; display: inline-block; height: 2.843ex; vertical-align: -0.838ex; width: 28.815ex;" /></span>.</li><li style="margin-bottom: 0.1em;">Si <i>A</i> et <i>B</i> sont <a href="https://fr.wikipedia.org/wiki/Ensemble_fini" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Ensemble fini">finis</a>, alors le <a class="mw-redirect" href="https://fr.wikipedia.org/wiki/Cardinal_d%27un_ensemble_fini" style="background: none; color: #3366cc; overflow-wrap: break-word; text-decoration-line: none;" title="Cardinal d'un ensemble fini">cardinal</a> de <i>A × B</i> est égal au produit des cardinaux de <i>A</i> et de <i>B</i></li></ul><div><a href="https://fr.wikipedia.org/wiki/Produit_cart%C3%A9sien">Produit cartésien — Wikipédia (wikipedia.org)</a></div><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;"><br /></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-4373788383727671702023-05-04T15:48:00.005+02:002023-05-04T16:16:53.130+02:00Factorisation des factorielles<p> </p><div align="center" style="background-color: #ffffcc;"><table border="1" cellpadding="0" cellspacing="1" class="MsoNormalTable" style="width: 836px;"><tbody><tr style="height: 114.55pt;"><td style="background: rgb(235, 255, 235); height: 114.55pt; padding: 3pt; width: 625.25pt;" width="834"><p align="center" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt; text-align: center;"><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 24pt;">FACTORISATION des FACTORIELLES<o:p></o:p></span></b></p><p align="center" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt; text-align: center;"><span style="font-size: 13.5pt;"> </span></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 53.35pt 0.0001pt 54.8pt; text-align: justify;"><b><span face="Arial, "sans-serif"" style="font-size: 14pt;">Méthode de <a href="http://villemin.gerard.free.fr/Wwwgvmm/Premier/Facto.htm" style="color: blue;">factorisation</a>.<o:p></o:p></span></b></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 53.35pt 0.0001pt 54.8pt; text-align: justify;"><b><span face="Arial, "sans-serif"" style="font-size: 14pt;">En déduire quelques caractéristiques, comme la quantité de <a href="http://villemin.gerard.free.fr/Wwwgvmm/Nombre/ZerArith.htm" style="color: blue;">zéros</a> à droite.</span></b><b><span face="Arial, "sans-serif""><o:p></o:p></span></b></p></td></tr></tbody></table></div><p align="center" style="background-color: #ffffcc; font-family: "Times New Roman", "serif"; margin: 0cm 0cm 0.0001pt; text-align: center;"><span style="font-size: 8pt;"> </span></p><p align="center" class="MsoNormal" style="background-color: #ffffcc; font-family: "Times New Roman", "serif"; margin: 0cm 0cm 0.0001pt; text-align: center;"><span style="font-size: 14pt;"> </span></p><div align="center" style="background-color: #ffffcc;"><table border="1" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="border-collapse: collapse; border: none;"><tbody><tr style="height: 34.3pt;"><td colspan="4" style="background: rgb(252, 224, 200); border-color: rgb(255, 153, 0) rgb(250, 191, 143) silver rgb(255, 153, 0); border-style: solid; border-width: 2.25pt 1pt 1pt 2.25pt; height: 34.3pt; padding: 4.25pt 3.5pt; width: 573.65pt;" width="765"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 21.55pt 0.0001pt 13.3pt;"><a name="approche"></a><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;">Approche</span></b><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;"><o:p></o:p></span></b></p></td><td style="background: rgb(243, 243, 243); border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: 2.25pt solid rgb(255, 153, 0); height: 34.3pt; padding: 4.25pt 3.5pt; width: 56pt;" width="75"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt; text-align: center;"><span face="Calibri, "sans-serif""><map name="MicrosoftOfficeMap0"><area coords="6, 24, 12, 24, 12, 44, 25, 44, 25, 24, 32, 24, 19, 6, 6, 24" href="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact.htm#top" shape="Polygon"></area></map><img border="0" height="52" src="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact_fichiers/image010.gif" usemap="#MicrosoftOfficeMap0" v:shapes="_x0000_s1029 _x0000_s1030 _x0000_s1031" width="38" /></span><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;"><o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" width="363"><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">Formons la factorisation première des nombres <a href="http://villemin.gerard.free.fr/Wwwgvmm/Compter/SixFact.htm" style="color: blue;">factoriels</a>.<o:p></o:p></span></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 7.75pt;"><b><span face="Arial, "sans-serif"">2! = <span class="GramE">1 .</span> 2<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 7.75pt;"><b><span face="Arial, "sans-serif"">3! = <span class="GramE">1 .</span> <span class="GramE">2 .</span> 3<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 7.75pt;"><b><span face="Arial, "sans-serif"">4! = <span class="GramE">1 .</span> <span class="GramE">2 .</span> <span class="GramE">3 .</span> 4<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 7.75pt;"><b><span face="Arial, "sans-serif"">5! = <span class="GramE">1 .</span> <span class="GramE">2 .</span> <span class="GramE">3 .</span> <span class="GramE">4 .</span> 5<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 7.75pt;"><b><span face="Arial, "sans-serif"">6! = <span class="GramE">1 .</span> <span class="GramE">2 .</span> <span class="GramE">3 .</span> <span class="GramE">4 .</span> <span class="GramE">5 .</span> 6<o:p></o:p></span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 2<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 6<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 24<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 120<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 720<o:p></o:p></span></b></p></td><td colspan="2" style="background: rgb(255, 255, 151); border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" valign="top" width="159"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">= 2<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">= <span class="GramE">2 .</span> 3<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">= <span class="GramE">2<sup>3</sup> .</span> 3<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">= <span class="GramE">2<sup>3</sup> .</span> <span class="GramE">3 .</span> 5<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">= <span class="GramE">2<sup>4</sup> .</span> <span class="GramE">3<sup>2</sup> .</span> 5<o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" width="363"><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">Examinons le cas de 6!<o:p></o:p></span></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">Quel est l'exposant du facteur 2<o:p></o:p></span></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">On effectue les divisions indiquées<o:p></o:p></span></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">Arrêt si le résultat est inférieur à 1<o:p></o:p></span></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">On prend les parties entières de ces résultats<o:p></o:p></span></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">On les additionne<o:p></o:p></span></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">6/2 = 3<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">6/2<sup>2</sup> = 1,5<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">6/2<sup>3</sup> = 0,75<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif""> </span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">=> 3<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">=> 1<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><u><span face="Arial, "sans-serif"">=> 0<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 4<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"" style="font-size: 11pt;">4 est l'exposant du facteur 2<o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" width="363"><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">Même opération avec 3, le nombre premier immédiatement après 2<o:p></o:p></span></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">6/3 = 2<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">6/3<sup>2</sup> = 0,66<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif""> </span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">=> 2<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><u><span face="Arial, "sans-serif"">=> 0<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 2<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"" style="font-size: 11pt;">2 est l'exposant du facteur 3<o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" width="363"><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">Et avec le 5, le nombre premier suivant<o:p></o:p></span></p><p style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 31.75pt; text-indent: -13.9pt;"><span face="Arial, "sans-serif"" style="font-size: 14pt;">Inutile d'aller plus loin avec le nombre premier 7, car la division donnera un résultat inférieur à 1<o:p></o:p></span></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">6/5 = 1<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">6/5<sup>2</sup> = 0,24<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif""> </span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">=> 1<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><u><span face="Arial, "sans-serif"">=> 0<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 10pt;"><b><span face="Arial, "sans-serif"">= 1<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"" style="font-size: 11pt;">1 est l'exposant du facteur 5<o:p></o:p></span></b></p></td></tr><tr height="0"><td style="border: none;" width="329"></td><td style="border: none;" width="155"></td><td style="border: none;" width="118"></td><td style="border: none;" width="71"></td><td style="border: none;" width="72"></td></tr></tbody></table></div>
x
<div><br /></div><div><table border="1" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="background-color: #ffffcc; border-collapse: collapse; border: none;"><tbody><tr style="height: 34.3pt;"><td style="background: rgb(252, 224, 200); border-color: rgb(255, 153, 0) white silver rgb(255, 153, 0); border-style: solid; border-width: 2.25pt 1pt 1pt 2.25pt; height: 34.3pt; padding: 0cm 3.5pt; width: 569.3pt;" width="759"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 21.55pt 0.0001pt 13.3pt;"><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;">Théorème<o:p></o:p></span></b></p></td><td style="background: rgb(243, 243, 243); border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: 2.25pt solid rgb(255, 153, 0); height: 34.3pt; padding: 0cm 3.5pt; width: 61.05pt;" width="81"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt; text-align: center;"><span face="Calibri, "sans-serif""><map name="MicrosoftOfficeMap1"><area coords="6, 24, 12, 24, 12, 44, 25, 44, 25, 24, 32, 24, 19, 6, 6, 24" href="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact.htm#top" shape="Polygon"></area></map><img border="0" height="52" src="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact_fichiers/image010.gif" usemap="#MicrosoftOfficeMap1" v:shapes="_x0000_s1032 _x0000_s1033 _x0000_s1034" width="38" /></span><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;"><o:p></o:p></span></b></p></td></tr><tr style="height: 127.25pt;"><td colspan="2" style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid silver; border-top: none; height: 127.25pt; padding: 0cm 3.5pt; width: 630.35pt;" width="840"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><span face="Arial, "sans-serif"" style="font-size: 16pt;"> </span></p><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><span face="Arial, "sans-serif"" style="font-size: 16pt;">Si <b>n</b> est un nombre entier supérieur ou égal à <b>1</b> et<o:p></o:p></span></p><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><span face="Arial, "sans-serif"" style="font-size: 16pt;">si <b>p</b> est un nombre premier,<o:p></o:p></span></p><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><span face="Arial, "sans-serif"" style="font-size: 16pt;">l'exposant de <b>p</b> dans la décomposition de <b>n!</b> en facteurs premiers<o:p></o:p></span></p><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><span face="Arial, "sans-serif"" lang="PT-BR" style="font-size: 16pt;">est égal à<o:p></o:p></span></p><p align="center" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 17.85pt; text-align: center;"><span face="Calibri, "sans-serif"" style="font-size: 11pt; line-height: 16.8667px; position: relative; top: 18pt;"><img height="44" src="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact_fichiers/image012.gif" v:shapes="_x0000_i1025" width="13" /></span><b><span face="Arial, "sans-serif"" lang="PT-BR" style="font-size: 16pt;"> = [ n/p] + [ n/p<sup>2</sup>] + [ n/p<sup>3</sup>] + [ n/p<sup>4</sup>] + …<o:p></o:p></span></b></p><p align="center" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 18.35pt 0.0001pt 17.85pt; text-align: center;"><span face="Arial, "sans-serif"" lang="EN-US"> </span></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 43.95pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Rappel de notation:</span></b><span face="Arial, "sans-serif""> Les crochets droits indiquent que l'on retient la partie entière du nombre.<o:p></o:p></span></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 43.95pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Exemple:</span></b><span face="Arial, "sans-serif""> [5,32] = 5</span></p></td></tr></tbody></table></div>
x<table border="1" cellpadding="0" cellspacing="0" class="MsoNormalTable" style="background-color: #ffffcc; border-collapse: collapse; border: none;"><tbody><tr style="height: 34.3pt;"><td colspan="4" style="background: rgb(252, 224, 200); border-color: rgb(255, 153, 0) rgb(250, 191, 143) silver rgb(255, 153, 0); border-style: solid; border-width: 2.25pt 1pt 1pt 2.25pt; height: 34.3pt; padding: 4.25pt 3.5pt; width: 573.65pt;" width="765"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 21.55pt 0.0001pt 13.3pt;"><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;">Exemple avec 10!<a name="fact10"></a>, 00! et 1000!</span></b><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;"><o:p></o:p></span></b></p></td><td style="background: rgb(243, 243, 243); border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: 2.25pt solid rgb(255, 153, 0); height: 34.3pt; padding: 4.25pt 3.5pt; width: 56pt;" width="75"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt; text-align: center;"><span face="Calibri, "sans-serif""><map name="MicrosoftOfficeMap2"><area coords="6, 24, 12, 24, 12, 44, 25, 44, 25, 24, 32, 24, 19, 6, 6, 24" href="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact.htm#top" shape="Polygon"></area></map><img border="0" height="52" src="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact_fichiers/image010.gif" usemap="#MicrosoftOfficeMap2" v:shapes="_x0000_s1035 _x0000_s1036 _x0000_s1037" width="38" /></span><b><span face="Verdana, "sans-serif"" style="color: #993300; font-size: 18pt;"><o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(213, 254, 255); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" width="363"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 16.6pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Méthode directe<o:p></o:p></span></b></p></td><td colspan="4" style="background: rgb(213, 254, 255); border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 357.55pt;" width="477"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 16.6pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Application du théorème<o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td rowspan="5" style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" valign="top" width="363"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> </span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> </span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif"">10! = <span class="GramE">1 .</span> <span class="GramE">2 .</span> <span class="GramE">3 .</span> <span class="GramE">4 .</span> <span class="GramE">5 .</span> <span class="GramE">6 .</span> <span class="GramE">7 .</span> <span class="GramE">8 .</span> <span class="GramE">9 .</span> 10<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> = <span class="GramE">2 .</span> <span class="GramE">3 .</span> 2<sup>2</sup>. <span class="GramE">5 .</span> <span class="GramE">2.3 .</span> <span class="GramE">7 .</span> <span class="GramE">2<sup>3</sup> .</span> <span class="GramE">3<sup>2</sup> .</span> 2.5<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> <o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> </span></b><b><span face="Arial, "sans-serif"" style="font-size: 16pt;">= <span class="GramE"><span style="color: red;">2<sup>8</sup> .</span></span><span style="color: red;"> <span class="GramE">3<sup>4</sup> .</span> <span class="GramE">5<sup>2</sup> .</span> 7</span></span></b><b><span face="Arial, "sans-serif""><o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> <o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> = 3 628 800</span></b><o:p></o:p></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/2 = 5<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/2<sup>2</sup> = 2,5<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/2<sup>3</sup> = 1,25<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif""> </span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">=> 5<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">=> 2<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><u><span face="Arial, "sans-serif"">=> 1<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">= 8<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><b><span face="Arial, "sans-serif"" style="color: red;">2<sup>8<o:p></o:p></sup></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/3 = 3,33<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/3<sup>2</sup> = 1,11<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif""> </span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">=> 3<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><u><span face="Arial, "sans-serif"">=> 1<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">= 4<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><b><span face="Arial, "sans-serif"" style="color: red;">3<sup>4</sup><o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/5 = 2<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif""> </span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><u><span face="Arial, "sans-serif"">=> 2<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">= 2<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><b><span face="Arial, "sans-serif"" style="color: red;">5<sup>2</sup><o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/7 = 1,42<o:p></o:p></span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">=> 1<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><b><span face="Arial, "sans-serif"" style="color: red;">7<sup>1</sup><o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">10/11 = 0,9<o:p></o:p></span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 19pt;"><b><span face="Arial, "sans-serif"">=> 0<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><span face="Arial, "sans-serif"" style="color: red; font-size: 10pt;">Fin</span><b><span face="Arial, "sans-serif"" style="color: red;"><o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(213, 254, 255); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" width="363"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 16.6pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Cas de 100 !<o:p></o:p></span></b></p></td><td colspan="4" style="background: rgb(213, 254, 255); border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 357.55pt;" width="477"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 16.6pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Exemple de calcul pour 2<o:p></o:p></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" valign="top" width="363"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt;"><b><span face="Arial, "sans-serif""> </span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt;"><b><span face="Arial, "sans-serif""> </span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 7.75pt 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif"">100! = <span class="GramE">2<sup>97</sup> .</span> <span class="GramE">3<sup>48</sup> .</span> <span class="GramE">5<sup>24</sup> .</span> 7<sup>16</sup> .11<sup>9</sup> . <span class="GramE">13<sup>7</sup> .</span> <span class="GramE">17<sup>5</sup> .</span> <span class="GramE">19<sup>5</sup> .</span> <span class="GramE">23<sup>4</sup> .</span> <span class="GramE">29<sup>3</sup> .</span> <span class="GramE">31<sup>3</sup> .</span> <span class="GramE">37<sup>2</sup> .</span> <span class="GramE">41<sup>2</sup> .</span> <span class="GramE">43<sup>2</sup> .</span> <span class="GramE">47<sup>2</sup> .</span> <span class="GramE">53 .</span> <span class="GramE">59 .</span> <span class="GramE">61 .</span> <span class="GramE">67 .</span> <span class="GramE">71 .</span> <span class="GramE">73 .</span> <span class="GramE">79 .</span> <span class="GramE">83 .</span> <span class="GramE">89 .</span> 97<sup><o:p></o:p></sup></span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">100/2 = 50<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">100/2<sup>2</sup> = 25<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">100/2<sup>3</sup> = 12,5<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">100/2<sup>4</sup> = 6,25<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">100/2<sup>5</sup> = 3,12<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">100/2<sup>6</sup> = 1,56<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">100/2<sup>7</sup> = 0,78<o:p></o:p></span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 50<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 25<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 12<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 6<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 3<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 1<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><u><span face="Arial, "sans-serif"">=> 0<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">= 97<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><b><span face="Arial, "sans-serif"" style="color: red;">2<sup>97<o:p></o:p></sup></span></b></p></td></tr><tr style="height: 33.25pt;"><td style="background: rgb(213, 254, 255); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 272.1pt;" width="363"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 16.6pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Cas de 1000 !<o:p></o:p></span></b></p></td><td colspan="4" style="background: rgb(213, 254, 255); border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 33.25pt; padding: 4.25pt 3.5pt; width: 357.55pt;" width="477"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 0cm 0.0001pt 16.6pt;"><b><span face="Arial, "sans-serif"" style="color: #993300;">Exemple de calcul pour 5<o:p></o:p></span></b></p></td></tr><tr style="height: 99.35pt;"><td style="background: rgb(255, 251, 247); border-bottom: 1pt solid silver; border-left: 2.25pt solid rgb(255, 153, 0); border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 99.35pt; padding: 4.25pt 3.5pt; width: 272.1pt;" valign="top" width="363"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 7.75pt 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif"">1000! => se termine par 249 zéros<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 7.75pt 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif""> </span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm 7.75pt 0.0001pt 18.85pt;"><b><span face="Arial, "sans-serif"">En effet, la puissance de 5 est 249 et combiné avec 249 fois le 2 (il y beaucoup plus de 2, bien sûr), le produit donne 10 (donc un zéro) à chaque fois.</span></b><o:p></o:p></p></td><td style="background: white; border-bottom: 1pt solid silver; border-image: initial; border-left: none; border-right: none; border-top: none; height: 99.35pt; padding: 4.25pt 3.5pt; width: 137.25pt;" valign="top" width="183"><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">1000/5 = 200<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">1000/5<sup>2</sup> = 40<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">1000/5<sup>3</sup> = 8<o:p></o:p></span></b></p><p align="right" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: right;"><b><span face="Arial, "sans-serif"">1000/5<sup>4</sup> = 1,6<o:p></o:p></span></b></p></td><td style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid rgb(250, 191, 143); border-top: none; height: 99.35pt; padding: 4.25pt 3.5pt; width: 101.25pt;" valign="top" width="135"><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 200<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 40<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">=> 8<o:p></o:p></span></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><u><span face="Arial, "sans-serif"">=> 1<o:p></o:p></span></u></b></p><p class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm;"><b><span face="Arial, "sans-serif"">= 249<o:p></o:p></span></b></p></td><td colspan="2" style="background: white; border-bottom: 1pt solid silver; border-left: none; border-right: 1pt solid silver; border-top: none; height: 99.35pt; padding: 4.25pt 3.5pt; width: 119.05pt;" width="159"><p align="center" class="MsoNormal" style="font-family: "Times New Roman", "serif"; font-size: 12pt; margin: 0cm -1.4pt 0.0001pt 0cm; text-align: center;"><b><span face="Arial, "sans-serif"" style="color: red;">5<sup>249</sup></span></b></p></td></tr></tbody></table>
x<div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNOJU21obOv5BeTdyfze5mpuVEb3oLa_eoeDHVHB-pytVAG-L_kX6p8NLHz-pyov-hWLDI44PB7mJ0y8y-ScMY9Z_1UXdhsAkAEvDtj6_LXMpZh3-5PrnhOH3maUxmyo6qXtVHh9TqY-VTk1tjEAlnpPIxLg0Ed_8ItlpNjyVO4ET_fbHLxu_82dU2/s507/20.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="505" data-original-width="507" height="499" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEjNOJU21obOv5BeTdyfze5mpuVEb3oLa_eoeDHVHB-pytVAG-L_kX6p8NLHz-pyov-hWLDI44PB7mJ0y8y-ScMY9Z_1UXdhsAkAEvDtj6_LXMpZh3-5PrnhOH3maUxmyo6qXtVHh9TqY-VTk1tjEAlnpPIxLg0Ed_8ItlpNjyVO4ET_fbHLxu_82dU2/w501-h499/20.jpg" width="501" /></a></div><a href="http://villemin.gerard.free.fr/Wwwgvmm/Compter/FactProp.htm">Factorielles, propriétés (free.fr)</a><div><br /></div><div><a href="http://villemin.gerard.free.fr/ThNbCo01/ThFoFact.htm">TFA et factorielles (free.fr)</a></div><div><br /></div><br /><div><br /><div class="separator" style="clear: both; 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text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://villemin.gerard.free.fr/Wwwgvmm/Compter/Factsomm.htm#general" target="_blank">Villemin</a><br /></div>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-13358869263617813662023-05-04T10:29:00.000+02:002023-05-06T11:09:52.305+02:00Développement décimal de l'inverse d'un nombre premier<p> <span face="sans-serif" style="background-color: white; color: #252525; font-size: 14px;">En mathématiques, la période du</span><span face="sans-serif" style="background-color: white; color: #252525; font-size: 14px;"> </span><b style="color: #252525; font-family: sans-serif; font-size: 14px;">développement décimal périodique d’un nombre rationnel</b><span face="sans-serif" style="background-color: white; color: #252525; font-size: 14px;"> </span><span face="sans-serif" style="background-color: white; color: #252525; font-size: 14px;">est le cycle composé d’une séquence finie de chiffres qui se répète à l’infini.</span></p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">L'inverse d'un <a class="wml" href="https://wikimonde.com/article/Nombre_premier" style="background: none; color: #0b0080; text-decoration-line: none;" title="Nombre premier">nombre premier</a>, noté 1/p possède un <a class="wml" href="https://wikimonde.com/article/D%C3%A9veloppement_d%C3%A9cimal_p%C3%A9riodique" style="background: none; color: #0b0080; text-decoration-line: none;" title="Développement décimal périodique">développement décimal périodique</a> dont la longueur de la période est notée δ<sub style="line-height: 1;">(p)</sub></p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Ceci exclut les nombres premiers 2 et 5 dont l'inverse ne possède pas de développement décimal périodique</p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Exemple :</p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">1/7 = 0,<b>142857</b> 142857 142857...</p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">δ<sub style="line-height: 1;">(7)</sub> = 6</p><h3 style="background: none rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span class="mw-headline" id="Tableau_des_nombres_premiers_dont_.CE.B4p_.3C101">ableau des nombres premiers dont δ<sub style="line-height: 1;">p</sub> <101</span></h3><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: #222222; font-family: sans-serif; font-size: 14px; margin: 1em 0px;"><caption style="font-weight: bold;">Table des nombres premiers en fonction de la longueur de la période du développement décimal de leur inverse</caption><tbody><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Longueur de la période</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Nombres premiers</th></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">4</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">101</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41, 271</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">6</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7, 13</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">239, 4649</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">8</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">73, 137</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">9</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">333667</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">9091</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">21649, 513239</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">12</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">9901</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">13</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">53, 79, 265371653</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">14</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">909091</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">15</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">31, 2906161</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">16</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">17, 5882353</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">17</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2071723, 5363222357</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">18</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">19, 52579</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">19</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1111111111111111111</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">20</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3541, 27961</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">21</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">43, 1933, 10838689</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">22</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">23, 4093, 8779</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">23</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11111111111111111111111</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">24</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">99990001</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">25</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">21401, 25601, 182521213001</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">26</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">859, 1058313049</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">27</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">757, 440334654777631</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">28</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">29, 281, 121499449</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">29</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3191, 16763, 43037, 62003, 77843839397</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">30</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">211, 241, 2161</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">31</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2791, 6943319, 57336415063790604359</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">32</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">353, 449, 641, 1409, 69857</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">33</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">67, 1344628210313298373</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">34</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">103, 4013, 21993833369</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">35</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">71, 123551, 102598800232111471</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">36</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">999999000001</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2028119, 247629013, 2212394296770203368013</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">38</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">909090909090909091</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">39</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">900900900900990990990991</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">40</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1676321, 5964848081</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">83, 1231, 538987, 201763709900322803748657942361</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">42</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">127, 2689, 459691</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">43</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">173, 1527791, 1963506722254397, 2140992015395526641</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">44</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">89, 1052788969, 1056689261</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">45</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">238681, 4185502830133110721</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">46</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">47, 139, 2531, 549797184491917</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">47</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">35121409, 316362908763458525001406154038726382279</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">48</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">9999999900000001</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">49</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">505885997, 1976730144598190963568023014679333</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">50</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">251, 5051, 78875943472201</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">51</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">613, 210631, 52986961, 13168164561429877</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">52</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">521, 265371653, 1900381976777332243781</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">53</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">107, 1659431, 1325815267337711173, 47198858799491425660200071</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">54</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">70541929, 14175966169</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">55</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1321, 62921, 83251631, 1300635692678058358830121</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">56</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7841, 127522001020150503761</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">57</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">21319, 10749631, 3931123022305129377976519</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">58</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">59, 154083204930662557781201849</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">59</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2559647034361, 4340876285657460212144534289928559826755746751</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">60</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">61, 4188901, 39526741</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">61</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">733, 4637, 329401 , 974293 , 1360682471 , 106007173861643 , 7061709990156159479</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">62</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">909090909090909090909090909091</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">63</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10837, 23311, 45613, 45121231, 1921436048294281</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">64</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">19841, 976193, 6187457, 834427406578561</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">65</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">162503518711, 5538396997364024056286510640780600481</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">66</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">599144041, 183411838171</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">67</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">493121, 79863595778924342083, 28213380943176667001263153660999177245677</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">68</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">28559389, 1491383821, 2324557465671829</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">69</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">277, 203864078068831, 1595352086329224644348978893</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">70</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">4147571, 265212793249617641</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">71</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">241573142393627673576957439049, 45994811347886846310221728895223034301839</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">72</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3169, 98641, 3199044596370769</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">73</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">12171337159 , 1855193842151350117, 49207341634646326934001739482502131487446637</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">74</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7253, 422650073734453, 296557347313446299</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">75</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">151, 4201, 15763985553739191709164170940063151</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">76</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">722817036322379041, 1369778187490592461</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">77</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5237, 42043, 29920507, 136614668576002329371496447555915740910181043</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">78</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">157, 6397, 216451, 388847808493</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">79</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">317, 6163, 10271, 307627, 49172195536083790769 , 3660574762725521461527140564875080461079917</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">80</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5070721, 19721061166646717498359681</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">81</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">163, 9397, 2462401, 676421558270641, 130654897808007778425046117</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">82</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2670502781396266997, 3404193829806058997303</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">83</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3367147378267 , 9512538508624154373682136329, 346895716385857804544741137394505425384477</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">84</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">226549, 4458192223320340849</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">85</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">262533041, 8119594779271, 4222100119405530170179331190291488789678081</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">86</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">57009401, 2182600451, 7306116556571817748755241</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">87</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">4003, 72559, 310170251658029759045157793237339498342763245483</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">88</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">617, 16205834846012967584927082656402106953</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">89</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">497867, 103733951 , 104984505733 , 5078554966026315671444089 , 403513310222809053284932818475878953159</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">90</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">29611, 3762091, 8985695684401</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">91</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">547, 14197, 17837, 4262077, 43442141653, 316877365766624209 , 110742186470530054291318013</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">92</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1289, 18371524594609 , 4181003300071669867932658901</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">93</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">900900900900900900900900900900990990990990990990990990990991</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">94</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">6299, 4855067598095567, 297262705009139006771611927</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">95</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">191, 59281, 63841, 1289981231950849543985493631, 965194617121640791456070347951751</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">96</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">97, 206209, 66554101249, 75118313082913</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">97</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">12004721, 846035731396919233767211537899097169, 109399846855370537540339266842070119107662296580348039</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">98</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">197, 5076141624365532994918781726395939035533</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">99</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">199, 397, 34849, 362853724342990469324766235474268869786311886053883</td></tr><tr><td scope="row" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">100</td><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">60101 , 7019801, 182521213001 , 14103673319201 , 1680588011350901</td></tr></tbody></table><h2 style="background: none rgb(255, 255, 255); border-bottom: 1px solid rgb(162, 169, 177); font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span id="Factorisation_de_1_+_10_+_102_+_..._+_10k-1"></span><span class="mw-headline" id="Factorisation_de_1_.2B_10_.2B_102_.2B_..._.2B_10k-1">Factorisation de 1 + 10 + 10<sup style="line-height: 1;">2</sup> + ... + 10<sup style="line-height: 1;">k-1</sup></span></h2><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Le principe de cette classification est lié à la factorisation de 1 + 10 + 10<sup style="line-height: 1;">2</sup> + ... + 10<sup style="line-height: 1;">k-1</sup>. En effet, soit <i>p</i> un nombre premier dont la longueur de période est <i>k</i>, l'inverse de <i>p</i> a pour développement décimal <span class="error" style="color: #dd3333; font-size: larger;">Argument vide ou contenant un symbole « = » à mettre entre accolades doubles.</span> En multipliant l'égalité par <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">10<sup style="line-height: 1;"><i>k</i></sup> - 1</span>, on obtient le développement décimal de <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><span style="display: inline-block; font-size: 14.042px; text-align: center; vertical-align: -0.5em;"><span style="display: block; line-height: 1em; margin: 0px 0.1em;">10<sup style="line-height: 1;"><i>k</i></sup> - 1</span><span style="height: 1px; left: -10000px; overflow: hidden; position: absolute; top: auto; width: 1px;">/</span><span style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;"><i>p</i></span></span></span> <span class="error" style="color: #dd3333; font-size: larger;">Argument vide ou contenant un symbole « = » à mettre entre accolades doubles.</span> C'est un entier. Donc <i>p</i> divise <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">10<sup style="line-height: 1;"><i>k</i></sup> - 1</span>. Comme <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">10<sup style="line-height: 1;"><i>k</i></sup> - 1 = 9 × (1 + 10 + 10<sup style="line-height: 1;">2</sup> + ... + 10<sup style="line-height: 1;"><i>k</i>-1</sup>)</span>, pour <i>p</i> différent de 3, <i>p</i> doit être un diviseur de 1 + 10 + 10<sup style="line-height: 1;">2</sup> + ... + 10<sup style="line-height: 1;"><i>k</i>-1</sup>.</p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Soit N<sub style="line-height: 1;">1</sub>=1, N<sub style="line-height: 1;">2</sub>=11, N<sub style="line-height: 1;">3</sub>=111, ..., N<sub style="line-height: 1;">k</sub>=<math> \sum_{i=0}^{k-1}10^i </math> = 1 + 10 + 10<sup style="line-height: 1;">2</sup> + 10<sup style="line-height: 1;">3</sup> + ... + 10<sup style="line-height: 1;">k-1</sup> pour <math> k \in \N^* </math></p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">La factorisation de N<sub style="line-height: 1;">k</sub> peut se décomposer comme le produit de P × K × F<span class="reference" id="cite_ref-1" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://plus.wikimonde.com/wiki/D%C3%A9veloppement_d%C3%A9cimal_p%C3%A9riodique_de_l%27inverse_d%27un_nombre_premier#cite_note-1" style="background: none; color: #0b0080; text-decoration-line: none;">[1]</a></span> où</p><ul style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3Csvg xmlns=%22http://www.w3.org/2000/svg%22 version=%221.1%22 width=%225%22 height=%2213%22%3E %3Ccircle cx=%222.5%22 cy=%229.5%22 r=%222.5%22 fill=%22%2300528c%22/%3E %3C/svg%3E"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">P : produit des nombres premiers p dont δ<sub style="line-height: 1;">p</sub>=k</li><li style="margin-bottom: 0.1em;">K : produit des nombres premiers p dont δ<sub style="line-height: 1;">p</sub> est un <a class="wml" href="https://wikimonde.com/article/Diviseur" style="background: none; color: #0b0080; text-decoration-line: none;" title="Diviseur">diviseur</a> de k</li><li style="margin-bottom: 0.1em;">F : produits des nombres p<sup style="line-height: 1;">q</sup> ou p est un nombre premier et q un exposant >0 <math> \in \N </math> tel que :<ul style="list-style-image: url("data:image/svg+xml,%3Csvg xmlns=%22http://www.w3.org/2000/svg%22 version=%221.1%22 width=%225%22 height=%2213%22%3E %3Ccircle cx=%222.5%22 cy=%229.5%22 r=%222.5%22 fill=%22%2300528c%22/%3E %3C/svg%3E"); list-style-type: disc; margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">p<sup style="line-height: 1;">q</sup> soit un diviseur de k</li><li style="margin-bottom: 0.1em;">k ≥ δ <sub style="line-height: 1;">p</sub> × p</li></ul></li></ul><div class="mw-references-wrap" style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px;"><ol class="references" style="font-size: 11.9px; list-style-image: none; margin: 0.3em 0px 0px 3.2em; padding: 0px;"><li id="cite_note-1" style="margin-bottom: 0.1em;"><span class="mw-cite-backlink" style="user-select: none;"><a href="https://plus.wikimonde.com/wiki/D%C3%A9veloppement_d%C3%A9cimal_p%C3%A9riodique_de_l%27inverse_d%27un_nombre_premier#cite_ref-1" style="background: none; color: #0b0080; text-decoration-line: none;"><span class="cite-accessibility-label" style="border: 0px; clip: rect(1px, 1px, 1px, 1px); height: 1px; overflow: hidden; padding: 0px; position: absolute; top: -99999px; user-select: none; width: 1px;">Aller</span>↑</a></span> <span class="reference-text">calculé pour k<101</span></li></ol></div><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;"><b>Exemple pour k=6 :</b></p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Factorisation de 111111 (1+10+10<sup style="line-height: 1;">2</sup>+10<sup style="line-height: 1;">3</sup>+10<sup style="line-height: 1;">4</sup>+10<sup style="line-height: 1;">5</sup>)</p><ul style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3Csvg xmlns=%22http://www.w3.org/2000/svg%22 version=%221.1%22 width=%225%22 height=%2213%22%3E %3Ccircle cx=%222.5%22 cy=%229.5%22 r=%222.5%22 fill=%22%2300528c%22/%3E %3C/svg%3E"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">P = 7 × 13</li><li style="margin-bottom: 0.1em;">K = 11 × 37</li><li style="margin-bottom: 0.1em;">F = 3</li></ul><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">111111 = 7 × 13 × 11 × 37 × 3</p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;"><b>Exemple pour k = 78 :</b></p><p style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Factorisation de 111111111111111111111111111111111111111111111111111111111111111111111111111111</p><ul style="background-color: white; color: #252525; font-family: sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml,%3Csvg xmlns=%22http://www.w3.org/2000/svg%22 version=%221.1%22 width=%225%22 height=%2213%22%3E %3Ccircle cx=%222.5%22 cy=%229.5%22 r=%222.5%22 fill=%22%2300528c%22/%3E %3C/svg%3E"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">P = 157 × 6397 × 216451 × 388847808493</li><li style="margin-bottom: 0.1em;">K = 11x37x7x13x53x79x265371653x859x1058313049x900900900900990990990991</li><li style="margin-bottom: 0.1em;">F = 3 × 13</li></ul><h3 style="background: none rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span class="mw-headline" id="Calcul_des_facteurs_K_et_F">Calcul des facteurs K et F</span></h3><table class="wikitable" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: #222222; font-family: sans-serif; font-size: 14px; margin: 1em 0px;"><caption style="font-weight: bold;"></caption><tbody><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">k</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Diviseur de k</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Equivalent en nombre premier p (δ<sub style="line-height: 1;">(p)</sub> est un diviseur de k)</th><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">Coef. F</th></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">1</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">2</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">3</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">4</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">5</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">6</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">7</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">8</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">9</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">2</sup></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">10</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-5</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x41x271</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">11</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">12</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-6</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x7x13</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">13</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">14</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-7</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x239x4649</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">15</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-5</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x41x271</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">16</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-8</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x73x137</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">17</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">18</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-6-9</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x7x13x333667</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">2</sup></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">19</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">20</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-5-10</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x41x271x9091</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">21</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-7</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x239x4649</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">22</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-11</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x21649x513239</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">23</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">24</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-6-8-12</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x7x13</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">25</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41x271</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">26</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-13</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x53x79x265371653</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">27</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-9</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x333667</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">3</sup></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">28</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-7-14</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x239x4649x909091</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">29</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">30</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-5-6-10-15</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x41x271x7x13x9091x31x2906161</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">31</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">32</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-8-16</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x73x137x17x5882353</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">33</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-11</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x21649x513239</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">34</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-17</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x2071723x5363222357</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">35</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5-7</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41x271x239x4649</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">36</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-6-9-12-18</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x7x13x333667x9901x19x52579</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">2</sup></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">37</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">38</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-19</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x1111111111111111111</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">39</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-13</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x53x79x265371653</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">40</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-5-8-10-20</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x41x271x73x137x9091x3541x27961</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">41</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">42</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-6-7-14-21</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x7x13x239x4649x909091x43x1933x10838689</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3X7</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">43</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">44</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-11-22</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x21649x513239x23x4093x8779</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">45</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-5-9-15</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x41x271x333667x31x2906161</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">2</sup></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">46</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-23</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x11111111111111111111111</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">47</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">48</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-6-8-12-16-24</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x7x13x73x137x9901x17x5882353x99990001</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">49</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">239x4649</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">50</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-5-10-25</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x41x271x9091x21401x25601x182521213001</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">51</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-17</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x2071723x5363222357</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">52</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-13-26</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x53x79x265371653x859x1058313049</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">53</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">54</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-9-27</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x333667x757x440334654777631</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">3</sup></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">55</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5-11</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41x271x21649x513239</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">56</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-7-8-14-28</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x239x4649x73x137x909091x29x281x121499449</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">57</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-19</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x1111111111111111111</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">58</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-29</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x3191x16763x43037x62003x77843839397</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">59</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">60</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-5-6-10-12-15-20-30</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x41x271x7x13x9091x9901x31x2906161x3541x27961x211x241x2161</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">61</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">62</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-31</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x2791x6943319x57336415063790604359</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">63</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-7-9-21</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x239x4649x333667x43x1933x10838689</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">2</sup></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">64</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-8-16-32</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x73x137x17x5882353x353x449x641x1409x69857</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">65</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5-13</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41x271x9901</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">66</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-6-11-22-33</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x7x13x21649x513239x23x4093x8779x67x1344628210313298373</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3X11</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">67</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">68</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-17-34</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x207123x5363222357x103x4013x21993833369</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">69</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-23</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x11111111111111111111111</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">70</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-5-7-10-14-35</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x41x271x239x4649x9091x909091x71x123551x102598800232111471</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">71</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">72</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-6-8-9-12-18-24-36</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x7x13x73x137x333667x9901x19x52579x99990001x999999000001</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">2</sup></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">73</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">74</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-37</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1x2028119x247629013x2212394296770203368013</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">75</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-5-15-25</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37X41X271X31X2906161X99990001</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">76</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-19-38</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x1111111111111111111x909090909090909091</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">77</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7-11</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">239x4649x21649x513239</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">78</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-6-13-26-39</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x7x13x53x79x265371653x859x1058313049x900900900900990990990991</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3X13</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">79</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">80</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-5-8-10-16-20-40</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11X101X41X271X73X137X9091X17X5882353X3541X27961X1676321X5964848081</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">81</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-9-27</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x333667x757x440334654777631</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3<sup style="line-height: 1;">4</sup></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">82</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-41</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x83x1231x538987x201763709900322803748657942361</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">83</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">84</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-6-7-12-14-21-28-42</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x7x13x239x4649x9901x909091x43x1933x10838689x29x281x121499449x127x2689x459691</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3*7</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">85</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5-17</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41x271x2071723x5363222357</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">86</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-43</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x173x1527791x1963506722254397x2140992015395526641</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">87</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-29</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x3191x16763x43037x62003x77843839397</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">88</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-8-11-22-44</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x73x137x21649x513239x23x4093x8779x89x1052788969x1056689261</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">89</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">90</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-5-6-9-10-15-18-30-45</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x41x271x7x13x333667x9091x31x2906161x19x52579x211x241x2161x238681x4185502830133110721</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">91</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7-13</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">239x4649x53x79x265371653</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">92</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-23-46</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x11111111111111111111111x47x139x2531x549797184491917</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">93</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-31</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x2791x6943319x57336415063790604359</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">94</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-47</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x35121409x316362908763458525001406154038726382279</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">95</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5-19</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">41x271x1111111111111111111</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">96</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-3-4-6-8-12-16-24-32-48</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x37x101x7x13x73x137x9901x17x5882353x99990001x353x449x641x1409x69857x9999999900000001</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr bgcolor="F2F2F2"><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">97</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">98</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-7-14-49</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x239x4649x909091x505885997x1976730144598190963568023014679333</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;"></td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">99</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3-9-11-33</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">37x333667x21649x513239x67x1344628210313298373</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td></tr><tr><th style="background-color: #eaecf0; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;">100</th><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2-4-5-10-20-25-50</td><td bgcolor="#FFFF99" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11x101x41x251x271x9091x3541x5051x27961x21401x25601x182521213001x78875943472201</td></tr></tbody></table>
x<h3 style="background-color: white; color: #1b1b21; font-family: sans-serif;"><span class="mw-headline" id=".C3.89criture_fractionnaire_d.27un_d.C3.A9veloppement_p.C3.A9riodique">Écriture fractionnaire d'un développement périodique</span></h3><p style="background-color: white; color: #1b1b21; font-family: sans-serif; font-size: 13.2px; margin: 0px; padding: 0px;">Pour le développement périodique d'un nombre plus petit que 1, lorsque la période commence immédiatement après la virgule, la technique consiste à multiplier le nombre par la bonne puissance de 10 permettant de décaler complètement la période avant la virgule. Une soustraction permet alors de faire disparaître la partie décimale.</p><p style="background-color: white; color: #1b1b21; font-family: sans-serif; font-size: 13.2px; margin: 0px; padding: 0px;">Exemple :</p><dl style="background-color: white; color: #1b1b21; font-family: sans-serif; font-size: 13.2px;"><dd><img alt="x=0,2121\cdots=0,\underline{21}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/0/c/7/0c7d322b2c576114e7e0282f93141685.png" style="border: none; vertical-align: middle;" /></dd><dd><img alt=" 100x=21,2121\cdots =21,\underline{21}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/8/6/f/86f0b8b1f04fd90b4a9fe9b7c63867a5.png" style="border: none; vertical-align: middle;" /></dd><dd><img alt="100x-x = 21 \Leftrightarrow x=\frac{21}{99}=\frac{7}{33}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/d/d/1/dd15009cc47ce359ee0695f7de2928f4.png" style="border: none; vertical-align: middle;" /></dd></dl><p style="background-color: white; color: #1b1b21; font-family: sans-serif; font-size: 13.2px; margin: 0px; padding: 0px;">Si la période ne commence pas juste après la virgule, il faut commencer par multiplier le nombre par la bonne puissance de 10 pour faire démarrer le développement décimal périodique juste après la virgule, puis on utilise la méthode précédente sur la partie décimale.</p><p style="background-color: white; color: #1b1b21; font-family: sans-serif; font-size: 13.2px; margin: 0px; padding: 0px;">Exemple :</p><dl style="background-color: white; color: #1b1b21; font-family: sans-serif; font-size: 13.2px;"><dd><img alt="x=3,52121\cdots=3,5\underline{21}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/3/2/e/32e55a7795f92a5f29636fee77526333.png" style="border: none; vertical-align: middle;" /></dd><dd><img alt=" 10x=35,2121\cdots =35+0,\underline{21}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/9/7/c/97c803ac3388dfcb4400d445ed2cbd5c.png" style="border: none; vertical-align: middle;" /></dd><dd><img alt="10x= 35+\frac{7}{33}= \frac{1162}{33}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/f/c/1/fc1955d5a98d038d29062eb5e8e67205.png" style="border: none; vertical-align: middle;" /></dd><dd><img alt="x= \frac{1162}{330}= \dfrac{581}{165}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/5/a/9/5a9f0bb25a0893339034b0058f27637b.png" style="border: none; vertical-align: middle;" /></dd></dl><p style="background-color: white; color: #1b1b21; font-family: sans-serif; font-size: 13.2px; margin: 0px; padding: 0px;">Cet algorithme se généralise et conduit au résultat suivant :</p><div class="theoreme" style="background-color: white; border: 1px solid rgb(170, 170, 170); color: #1b1b21; font-family: sans-serif; font-size: 13.2px; margin: 1em 2em; padding: 0.5em 1em 0.4em; text-align: justify;"><strong class="theoreme-nom">Caractérisation des rationnels</strong><span class="theoreme-tiret"> — </span>Tout développement périodique est associé à un rationnel. En particulier, le rationnel associé au développement <img alt="0,\underline{a_1a_2\cdots a_{\ell}}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/b/1/9/b19415cb7c5ba5fc6f9d878d75cf4308.png" style="border: none; vertical-align: middle;" /> peut s'écrire<center><img alt="0,\underline{a_1a_2\cdots a_{\ell}}=\frac{a_1a_2\cdots a_{\ell}}{\underbrace{99\cdots 9}_{\ell\text{ chiffres}}}" class="tex" src="https://dictionnaire.sensagent.leparisien.fr/wiki-images-all/s/6/a/6/6a6c45f0d5a195df9169cc2b025b5858.png" style="border: none; vertical-align: middle;" /></center></div>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-46720850776571008232023-05-02T11:26:00.002+02:002023-05-02T11:33:24.768+02:00Zeckendorf<p> </p><div class="separator" style="clear: both; text-align: right;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwNlUqCsv1qrqw35ok3Fe-7Ac3D3bJNaly65HTBvyoFZLme2Tn6P2HmyCt-Pir3Of4fYB86cFk-hH3dgD3GHQSaQdBjUcTbtUwS8ljMJ56y0xQtHUQ_gk8QAQNB99CmY9vnbd8W8raxWPB_WoQHid2IqwIRfFJYtoEl7yDtmdIdPS5So-HG8X4P4fR/s759/zec1.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="616" data-original-width="759" height="321" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgwNlUqCsv1qrqw35ok3Fe-7Ac3D3bJNaly65HTBvyoFZLme2Tn6P2HmyCt-Pir3Of4fYB86cFk-hH3dgD3GHQSaQdBjUcTbtUwS8ljMJ56y0xQtHUQ_gk8QAQNB99CmY9vnbd8W8raxWPB_WoQHid2IqwIRfFJYtoEl7yDtmdIdPS5So-HG8X4P4fR/w395-h321/zec1.jpg" width="395" /></a></div><br /><div class="separator" style="clear: both; text-align: right;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvrRZqePdIaStCRCnAHuKRE0E_dsLu-BIAzRI9-USje_nk3Jvx-DMcv1BOzVRJFaOHHkj3U70loL8CxhfhXY4ifpc3UmJ7kWsrROQeoeeoZTK3UYiUF1u6Ofovmd4D7lO7le_m9bfu-sTFjB_4L16pXpFTphE-RN1SVoI6WDix1XhiG-Yk0zz_-INj/s764/zec.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="539" data-original-width="764" height="301" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgvrRZqePdIaStCRCnAHuKRE0E_dsLu-BIAzRI9-USje_nk3Jvx-DMcv1BOzVRJFaOHHkj3U70loL8CxhfhXY4ifpc3UmJ7kWsrROQeoeeoZTK3UYiUF1u6Ofovmd4D7lO7le_m9bfu-sTFjB_4L16pXpFTphE-RN1SVoI6WDix1XhiG-Yk0zz_-INj/w426-h301/zec.jpg" width="426" /></a></div><br /><span face="sans-serif" style="color: #594471;"><br /></span><p></p><img alt="exemple où le nombre mystère est 48" class="centered_image" height="337" src="https://sorciersdesalem.math.cnrs.fr/Fibonacci/exemple48b.png" style="color: #594471; display: block; font-family: sans-serif; margin: 20px auto; max-width: 100%;" title="Comment retrouver le nombre mystère" width="249" /><p style="color: #594471; font-family: sans-serif; margin-left: 30px; margin-right: 30px; margin-top: 0px;"><a href="https://sorciersdesalem.math.cnrs.fr/Fibonacci/explications_cartes_fibonacci.html">Magie des nombres de Fibonacci et décomposition de Zeckendorf (cnrs.fr)</a></p><p style="color: #594471; font-family: sans-serif; margin-left: 30px; margin-right: 30px; margin-top: 0px;"><br /></p><p style="color: #594471; font-family: sans-serif; margin-left: 30px; margin-right: 30px; margin-top: 0px;"><br /></p><p style="color: #594471; font-family: sans-serif; margin-left: 30px; margin-right: 30px; margin-top: 0px;"><a href="https://en.wikipedia.org/wiki/Zeckendorf%27s_theorem">Zeckendorf's theorem - Wikipedia</a></p><p style="color: #594471; font-family: sans-serif; margin-left: 30px; margin-right: 30px; margin-top: 0px;"><br /></p><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Le <dfn style="font-style: normal; font-weight: 700;">théorème de Zeckendorf</dfn>, dénommé ainsi d'après le <a href="https://wikimonde.com/article/Math%C3%A9maticien" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Mathématicien">mathématicien</a> <a href="https://wikimonde.com/article/Belgique" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Belgique">belge</a> <a href="https://wikimonde.com/article/%C3%89douard_Zeckendorf" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Édouard Zeckendorf">Édouard Zeckendorf</a>, est un <a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Théorème">théorème</a> de <a href="https://wikimonde.com/article/Th%C3%A9orie_additive_des_nombres" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Théorie additive des nombres">théorie additive des nombres</a> qui garantit que tout <a href="https://wikimonde.com/article/Entier_naturel" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Entier naturel">entier naturel</a> <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> peut être représenté, de manière unique, comme somme de <a class="mw-redirect" href="https://wikimonde.com/article/Nombre_de_Fibonacci" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Nombre de Fibonacci">nombres de Fibonacci</a> distincts et non consécutifs. Cette représentation est appelée la <b>représentation de Zeckendorf</b> de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>.</p><div aria-labelledby="mw-toc-heading" class="toc" id="toc" role="navigation" style="background-color: #f3f7f9; border: 1px solid rgb(209, 233, 249); color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 13.3px; left: 1px; line-height: 1.1; margin: 0px; padding: 0px; position: absolute; top: 190px; width: 210px; zoom: 1;"><div class="m-toc-title" style="background-color: #2a7ebc; border-bottom: 1px solid rgb(42, 126, 188); color: white; font-size: 17.29px; font-weight: bold; margin: -1px; padding: 0.1em; text-align: center; word-break: break-word;">Théorème de Zeckendorf</div><input class="toctogglecheckbox" id="toctogglecheckbox" role="button" style="direction: ltr; display: none; opacity: 0; position: absolute;" type="checkbox" /><ul style="list-style-image: none; list-style-type: none; margin: 0.3em 0px; padding: 0px;"><li class="toclevel-1 tocsection-1" style="border-bottom: 1px solid rgb(209, 233, 249); font-weight: bold; margin-bottom: 0.1em; padding: 0.2em;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#%C3%89nonc%C3%A9_et_exemple" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 204.688px;">Énoncé et exemple</span></a><ul style="list-style-position: outside; list-style-type: none; list-style: outside url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAUAAAANCAIAAADuXjPfAAAABnRSTlMA/wD/AP83WBt9AAAAHklEQVR4AWP4jwrowWcI6oEgEBtIISNCfFT9mOYDACO/lbNIGC/yAAAAAElFTkSuQmCC") none; margin: 0px 0px 0px 1em; padding-bottom: 0px; padding-left: 4px !important; padding-right: 0px; padding-top: 0px; padding: 0px 0px 0px 4px;"><li class="toclevel-2 tocsection-2" style="font-weight: normal; margin-bottom: 0.1em; padding: 0px;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Note_historique" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 187.391px;">Note historique</span></a></li></ul></li><li class="toclevel-1 tocsection-3" style="border-bottom: 1px solid rgb(209, 233, 249); font-weight: bold; margin-bottom: 0.1em; padding: 0.2em;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#D%C3%A9monstration" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 204.688px;">Démonstration</span></a><ul style="list-style-position: outside; list-style-type: none; list-style: outside url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAUAAAANCAIAAADuXjPfAAAABnRSTlMA/wD/AP83WBt9AAAAHklEQVR4AWP4jwrowWcI6oEgEBtIISNCfFT9mOYDACO/lbNIGC/yAAAAAElFTkSuQmCC") none; margin: 0px 0px 0px 1em; padding-bottom: 0px; padding-left: 4px !important; padding-right: 0px; padding-top: 0px; padding: 0px 0px 0px 4px;"><li class="toclevel-2 tocsection-4" style="font-weight: normal; margin-bottom: 0.1em; padding: 0px;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Repr%C3%A9sentation_des_premiers_entiers" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 187.391px;">Représentation des premiers entiers</span></a></li></ul></li><li class="toclevel-1 tocsection-5" style="border-bottom: 1px solid rgb(209, 233, 249); font-weight: bold; margin-bottom: 0.1em; padding: 0.2em;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Variations" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 204.688px;">Variations</span></a><ul style="list-style-position: outside; list-style-type: none; list-style: outside url("data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAUAAAANCAIAAADuXjPfAAAABnRSTlMA/wD/AP83WBt9AAAAHklEQVR4AWP4jwrowWcI6oEgEBtIISNCfFT9mOYDACO/lbNIGC/yAAAAAElFTkSuQmCC") none; margin: 0px 0px 0px 1em; padding-bottom: 0px; padding-left: 4px !important; padding-right: 0px; padding-top: 0px; padding: 0px 0px 0px 4px;"><li class="toclevel-2 tocsection-6" style="font-weight: normal; margin-bottom: 0.1em; padding: 0px;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Repr%C3%A9sentation_par_des_nombres_de_Fibonacci_d'indices_n%C3%A9gatifs" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 187.391px;">Représentation par des nombres de Fibonacci d'indices négatifs</span></a></li><li class="toclevel-2 tocsection-7" style="font-weight: normal; margin-bottom: 0.1em; padding: 0px;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Multiplication_de_Fibonacci" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 187.391px;">Multiplication de Fibonacci</span></a></li><li class="toclevel-2 tocsection-8" style="font-weight: normal; margin-bottom: 0.1em; padding: 0px;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Autres_suites" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 187.391px;">Autres suites</span></a></li><li class="toclevel-2 tocsection-9" style="font-weight: normal; margin-bottom: 0.1em; padding: 0px;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Syst%C3%A8me_d'Ostrowski" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 187.391px;">Système d'Ostrowski</span></a></li></ul></li><li class="toclevel-1 tocsection-10" style="border-bottom: 1px solid rgb(209, 233, 249); font-weight: bold; margin-bottom: 0.1em; padding: 0.2em;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Notes_et_r%C3%A9f%C3%A9rences" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; text-decoration: inherit; vertical-align: top; width: 204.688px;">Notes et références</span></a></li><li class="toclevel-1 tocsection-11" style="border-bottom: 1px solid rgb(209, 233, 249); font-weight: bold; margin-bottom: 0.1em; padding: 0.2em;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#Voir_aussi" style="background: 0px 0px; color: #202889; text-decoration-line: none;"><span class="toctext" style="display: inline-block; 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font-size: 14px; margin: 1em 2em; padding: 0.5em 1em 0.4em; text-align: justify;"><p style="line-height: inherit; margin: 0.5em 0px;"><strong class="theoreme-nom">Théorème de Zeckendorf<span class="reference" id="cite_ref-Zeck_1-0" style="font-size: 0.8em; font-weight: 400; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-Zeck-1" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">1</a></span></strong><span class="theoreme-tiret"> — </span>Pour tout entier naturel <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>, il existe une unique suite d’entiers <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>c</i><sub style="line-height: 1;">0</sub>, ... , <i>c<sub style="line-height: 1;">k</sub></i></span>, avec <span class="mwe-math-element"><img alt="{\displaystyle c_{0}\geq 2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/108915d447f5d0bbddd122c47de04921fe691848" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 6.322ex;" /></span> et <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>c</i><sub style="line-height: 1;"><i>i</i>+1</sub> > <i>c<sub style="line-height: 1;">i</sub></i> + 1</span>, tels que</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle N=\sum _{i=0}^{k}F_{c_{i}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/169a342ce976392405e0d2193d922e519149ce56" style="border: 0px; height: 7.343ex; vertical-align: -3.005ex; width: 11.968ex;" /></span>,</dd></dl><p style="line-height: inherit; margin: 0.5em 0px;">où <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">n</sub></span> est le <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span>-ième nombre de Fibonacci.</p></div><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Par exemple, <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">0</span> est représenté par la <a href="https://wikimonde.com/article/Somme_vide" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Somme vide">somme vide</a>. La représentation de Zeckendorf du nombre <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">100</span> est</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 100=89+8+3}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8e9d5ed12089859c4b7cb14616e26663bd86ad1a" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 16.916ex;" /></span>.</dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Le nombre <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">100</span> possède d'autres représentations comme somme de nombres de Fibonacci. Ainsi :</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 100=89+8+2+1}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdfa4800f6f3675caa8f8cf9b18d7488f712f4b8" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 20.919ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 100=89+5+3+2+1}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/162fce6927cc53a84d2528b9f1e47de37937cf74" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 24.922ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 100=55+34+8+3}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b5c05126b845990356ce9986dfb495f12388a489" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 22.082ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 100=55+34+8+2+1}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60f310652801af3bb6b10ab33377f6c8f064a958" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 26.085ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 100=55+34+5+3+2+1}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e2552e286b8489f40116030133596b38dbf75813" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 30.087ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 100=55+21+13+8+3}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/494474ec460a8ecbf8de816cb0feaa6d7b660663" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 27.247ex;" /></span></dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">mais ces représentations contiennent des nombres de Fibonacci consécutifs. À toute représentation d'un entier <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>, on associe un <a href="https://wikimonde.com/article/Mot_%28math%C3%A9matiques%29" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Mot (mathématiques)">mot</a> binaire, dont la <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span>-ième lettre est <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">1</span> si <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">n</sub></span> figure dans la représentation de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> et <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">0</span> sinon. Ainsi, aux représentations de <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">100</span> ci-dessus sont associés les mots :</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 1000010100}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd512e1b1c95b7140b06d73e8bba3e876f1a9148" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 11.625ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 1000010011}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe977b8e040e93c5f134019ca01433b743080e5b" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 11.625ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 1000001111}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65015ca0aaa6f4f1d79c344ec9bc1b27654b1361" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 11.625ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 110010100}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a70bd3a32a49b47e6eee849fd5123a2a3c98f9bc" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 10.462ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 110010011}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c0a0e20ac54c85b8db8cbd4bf25d85572e19eb60" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 10.462ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 110001111}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b0f0dff488b3c770764ca3d0586de2c3406cb856" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 10.462ex;" /></span></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 101110100}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0e19bbde9e8e27876fd8e1198203cc84832185e0" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 10.462ex;" /></span>.</dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">L'ensemble des mots binaires associés aux représentations de Zeckendorf forme un <a href="https://wikimonde.com/article/Langage_rationnel" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Langage rationnel">langage rationnel</a> : ce sont le mot vide et les mots commençant par <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">1</span> et ne contenant pas deux <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">1</span> consécutifs. Une <a class="mw-redirect" href="https://wikimonde.com/article/Expression_rationnelle" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Expression rationnelle">expression rationnelle</a> de ce langage est</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle 0+1(0+01)^{*}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0199903e3ff0b9ed69ffef010a71c975b05cd5bc" style="border: 0px; height: 2.843ex; vertical-align: -0.838ex; width: 14.357ex;" /></span>.</dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Le <a href="https://wikimonde.com/article/Codage_de_Fibonacci" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Codage de Fibonacci">codage de Fibonacci</a> d'un entier est, par définition, le mot binaire associé à sa représentation, retourné et suivi d'un symbole <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">1</span>. Ainsi, le codage de Fibonacci du nombre <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">100</span> est <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">00101000011</span>.</p><h3 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: Georgia, Times, serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span class="mw-headline" id="Note_historique">Note historique</span></h3><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Zeckendorf a publié sa démonstration du théorème en 1972<span class="reference" id="cite_ref-Zeck_1-1" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-Zeck-1" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">1</a></span>, alors que l'énoncé était connu, sous le nom de « théorème de Zeckendorf », depuis longtemps. Ce paradoxe est expliqué dans l'introduction de l'article de Zeckendorf : un autre mathématicien, Gerrit Lekkerkerker , a rédigé la preuve du théorème (et d'autres résultats) à la suite d'un exposé de Zeckendorf, et l'a publié<span class="reference" id="cite_ref-Lekkerkerker_2-0" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-Lekkerkerker-2" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">2</a></span> en 1952, tout en attribuant la paternité à Zeckendorf. D'après <a href="https://wikimonde.com/article/Clark_Kimberling" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Clark Kimberling">Clark Kimberling</a><span class="reference" id="cite_ref-Kimberling_3-0" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-Kimberling-3" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">3</a></span>, c'est un article de David E. Daykin<span class="reference" id="cite_ref-Daykin_4-0" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-Daykin-4" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">4</a></span>, publié dans un journal prestigieux, qui a contribué à faire connaître le résultat et son auteur.</p><h2 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px solid rgb(162, 169, 177); font-family: Georgia, Times, serif; font-weight: 400; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span id="D.C3.A9monstration"></span><span class="mw-headline" id="Démonstration">Démonstration</span></h2><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">La preuve du théorème est en deux parties :</p><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">1. <b>Existence</b> : L'existence de la représentation se prouve par l'emploi de l'<a href="https://wikimonde.com/article/Algorithme_glouton" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Algorithme glouton">algorithme glouton</a> ou par récurrence sur <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>.</p><div class="NavFrame demonstration" style="background-color: white; border-collapse: collapse; border: thin solid rgb(170, 170, 170); color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin: 1em 2em; overflow: hidden; padding: 0px 1em; text-align: justify;"><div class="NavHead" style="background: transparent; font-weight: 700; padding: 0px;"><strong>Deux preuves de l'existence</strong></div><div class="NavContent" style="padding-bottom: 0.4em;"><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dt style="font-weight: 700; margin-bottom: 0.1em;">Première preuve</dt><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">On raisonne par <a href="https://wikimonde.com/article/Raisonnement_par_r%C3%A9currence#R%C3%A9currence_bien_fond%C3%A9e" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Raisonnement par récurrence">récurrence bien fondée</a> sur l'entier naturel <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>. Si <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> est nul, il est représenté par la somme vide. Sinon, soit <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">k</sub></span>, avec <span class="mwe-math-element"><img alt="{\displaystyle k\geq 2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c797a67c0a51167d373c013a9a020f4568a11754" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 5.472ex;" /></span>, le plus grand nombre de Fibonacci inférieur ou égal à <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> et soit <span class="mwe-math-element"><img alt="{\displaystyle M=N-F_{k}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f377224ca28b77ac0e5ce4db2ccd2a924fbb1a21" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 13.028ex;" /></span>. Par hypothèse de récurrence, <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">M</span> possède une représentation. Alors, pour tout nombre <span class="mwe-math-element"><img alt="{\displaystyle F_{\ell }}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7524db1fcca8bb76c2a493490b82d1947f2338ad" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 2.413ex;" /></span> de cette représentation, <span class="mwe-math-element"><img alt="{\displaystyle F_{k}+F_{\ell }\leq F_{k}+M=N<F_{k+1}=F_{k}+F_{k-1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2cb9780736fbb69197c40a42ffa85fe9a8ec8a28" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 44.951ex;" /></span> donc <span class="mwe-math-element"><img alt="{\displaystyle F_{\ell }<F_{k-1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f52c02f237ce4d6225715a9fb3f8c89f3ce5cd1" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 10.195ex;" /></span>. La représentation de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">M</span>, augmentée de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">k</sub></span>, est donc bien une représentation de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>.</dd><dt style="font-weight: 700; margin-bottom: 0.1em;">Seconde preuve</dt><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">On raisonne par récurrence sur l'entier <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span> considéré. L'existence est facilement vérifiable pour les petites valeurs de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span>. Supposons qu'elle soit vraie pour un entier <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span> donné et décomposons cet entier en rangeant les nombres de Fibonacci par ordre croissant. Plusieurs cas sont à considérer :<ul style="list-style-image: url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1IiBoZWlnaHQ9IjEzIj4gPGNpcmNsZSBjeD0iMi41IiBjeT0iOS41IiByPSIyLjUiIGZpbGw9IiMwMDUyOGMiLz4gPC9zdmc+"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">Si <span class="mwe-math-element"><img alt="{\displaystyle n=F_{i_{1}}+F_{i_{2}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db9ec928a5c360d603f5e13811a95d6fd85e9d3" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 25.074ex;" /></span> avec <span class="mwe-math-element"><img alt="{\displaystyle 4\leq i_{1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a142c67cc8d61ee9192face0979d31d3f1d471b4" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 6.118ex;" /></span> et <span class="mwe-math-element"><img alt="{\displaystyle \forall j,i_{j+1}\geq i_{j}+2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da8a00d71df510895cc0f8b81b61d70a3c99388" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 15.911ex;" /></span>, alors :<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle n+1=F_{2}+F_{i_{1}}+F_{i_{2}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/338940e024bda1f84d48ac4664200acf803bd3b2" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 34.466ex;" /></span></dd></dl></li><li style="margin-bottom: 0.1em;">Si <span class="mwe-math-element"><img alt="{\displaystyle n=F_{3}+F_{5}+\cdots +F_{2r-1}+F_{i_{r}}+F_{i_{r+1}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4bffbe36f8ca952b99884be46f7d77d3e4b3339c" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 51.222ex;" /></span> avec <span class="mwe-math-element"><img alt="{\displaystyle i_{r}\geq 2r+2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4dc4a79eb0af01ed725a294e3d43824faf6746db" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 11.089ex;" /></span> et <span class="mwe-math-element"><img alt="{\displaystyle \forall j,i_{j+1}\geq i_{j}+2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da8a00d71df510895cc0f8b81b61d70a3c99388" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 15.911ex;" /></span> (et éventuellement <span class="mwe-math-element"><img alt="{\displaystyle i_{s}=2r-1}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/639df872062dad1a17f7f65e4dadd3cf4b561547" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 11.118ex;" /></span> auquel cas les termes <span class="mwe-math-element"><img alt="{\displaystyle F_{i_{r}}+F_{i_{r+1}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ecf7aa3e84130a206888113853a74d64e9e705b" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 22.155ex;" /></span> n'existent pas). Alors :<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle n+1=F_{2r}+F_{i_{r}}+F_{i_{r+1}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a014cfa3cd1361fa8afec6cc21736ccbc07dca89" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 36.782ex;" /></span></dd></dl></li><li style="margin-bottom: 0.1em;">Si <span class="mwe-math-element"><img alt="{\displaystyle n=F_{2}+F_{4}+\cdots +F_{2r-2}+F_{i_{r}}+F_{i_{r+1}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a35da0eec215aa00b6ac0f00cfc3e6d913764c93" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 51.222ex;" /></span> avec <span class="mwe-math-element"><img alt="{\displaystyle i_{r}\geq 2r+1}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2f30c8af65158c93daf66ef6e08652676947a347" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 11.089ex;" /></span> et <span class="mwe-math-element"><img alt="{\displaystyle \forall j,i_{j+1}\geq i_{j}+2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da8a00d71df510895cc0f8b81b61d70a3c99388" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 15.911ex;" /></span> (et éventuellement <span class="mwe-math-element"><img alt="{\displaystyle i_{s}=2r-2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/40ac3c63f67e0063177888f6cb3806d30c3e6b44" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 11.118ex;" /></span> auquel cas les termes <span class="mwe-math-element"><img alt="{\displaystyle F_{i_{r}}+F_{i_{r+1}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4ecf7aa3e84130a206888113853a74d64e9e705b" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 22.155ex;" /></span> n'existent pas). Alors :<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle n+1=F_{2r-1}+F_{i_{r}}+F_{i_{r+1}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f6a3e6911e7b26c52078a108ecfcc4e1e59d41c2" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 38.883ex;" /></span></dd></dl></li></ul></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">Ainsi, <span class="mwe-math-element"><img alt="n+1" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a135e65a42f2d73cccbfc4569523996ca0036f1" style="border: 0px; height: 2.343ex; vertical-align: -0.505ex; width: 5.398ex;" /></span> admet aussi une décomposition.</dd></dl></div><div style="clear: both;"></div></div><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">2. <b>Unicité</b> : Pour cette partie, on utilise le lemme suivant :</p><div class="theoreme" style="background-color: white; border: 1px solid rgb(170, 170, 170); color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin: 1em 2em; padding: 0.5em 1em 0.4em; text-align: justify;"><p style="line-height: inherit; margin: 0.5em 0px;"><strong class="theoreme-nom">Lemme</strong><span class="theoreme-tiret"> — </span> La somme de tout ensemble non vide de nombres de Fibonacci distincts et non consécutifs, dont le plus grand élément est <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">j</sub></span>, est strictement inférieure à <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>F</i><sub style="line-height: 1;"><i>j</i>+1</sub></span>.</p></div><div class="NavFrame demonstration" style="background-color: white; border-collapse: collapse; border: thin solid rgb(170, 170, 170); color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin: 1em 2em; overflow: hidden; padding: 0px 1em; text-align: justify;"><div class="NavHead" style="background: transparent; font-weight: 700; padding: 0px;"><strong>Deux démonstrations du lemme</strong></div><div class="NavContent" style="padding-bottom: 0.4em;"><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dt style="font-weight: 700; margin-bottom: 0.1em;">Première démonstration</dt><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">On raisonne par <a href="https://wikimonde.com/article/Raisonnement_par_r%C3%A9currence#R%C3%A9currence_simple_sur_les_entiers" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Raisonnement par récurrence">récurrence simple</a> sur le nombre d'éléments d'un tel ensemble <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">S</span>. L'initialisation est immédiate. Pour l'hérédité, si <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">S</span> a plus d'un élément, enlevons <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">j</sub></span>. Par hypothèse de récurrence, la somme des éléments restants est strictement inférieure à <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>F</i><sub style="line-height: 1;"><i>j</i>-1</sub></span>, donc la somme totale est strictement inférieure à <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>F</i><sub style="line-height: 1;"><i>j</i>+1</sub> + <i>F<sub style="line-height: 1;">j</sub></i> = <i>F</i><sub style="line-height: 1;"><i>j</i>+1</sub></span>.</dd><dt style="font-weight: 700; margin-bottom: 0.1em;">Seconde démonstration</dt><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">Si <span class="mwe-math-element"><img alt="{\displaystyle n=F_{i_{1}}+F_{i_{2}}+\cdots +F_{i_{s}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3db9ec928a5c360d603f5e13811a95d6fd85e9d3" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 25.074ex;" /></span> avec <span class="mwe-math-element"><img alt="{\displaystyle \forall j,i_{j+1}\geq i_{j}+2}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0da8a00d71df510895cc0f8b81b61d70a3c99388" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 15.911ex;" /></span>. Alors :<ul style="list-style-image: url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1IiBoZWlnaHQ9IjEzIj4gPGNpcmNsZSBjeD0iMi41IiBjeT0iOS41IiByPSIyLjUiIGZpbGw9IiMwMDUyOGMiLz4gPC9zdmc+"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;">si <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">i<sub style="line-height: 1;">s</sub></span> est pair :<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle F_{i_{1}}+F_{i_{2}}+\cdots +F_{i_{s-1}}\leq F_{i_{s}-2}+F_{i_{s}-4}+\cdots +F_{2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/360a040fe594a9c8f1eb1300f4a23dc9c0f288c8" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 49.548ex;" /></span></dd></dl></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">donc :<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle F_{i_{1}}+F_{i_{2}}+\cdots +F_{i_{s-1}}<F_{i_{s}-2}+F_{i_{s}-4}+\cdots +F_{2}+1=F_{i_{s}-1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8109cf131af05483474796d3b43fdee956fbde36" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 61.834ex;" /></span></dd></dl></dd></dl></li><li style="margin-bottom: 0.1em;">si <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">i<sub style="line-height: 1;">s</sub></span> est impair :<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle F_{i_{1}}+F_{i_{2}}+\cdots +F_{i_{s-1}}\leq F_{i_{s}-2}+F_{i_{s}-4}+\cdots +F_{3}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fd6fe98f6d5d0e2b7c6580fd0ef5e05b51a521f" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 49.548ex;" /></span></dd></dl></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">donc :<dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle F_{i_{1}}+F_{i_{2}}+\cdots +F_{i_{s-1}}<F_{i_{s}-2}+F_{i_{s}-4}+\cdots +F_{3}+1=F_{i_{s}-1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/984d80d8c37efd6e1ebe3ada98f3610c544dc52d" style="border: 0px; height: 2.843ex; vertical-align: -1.005ex; width: 61.834ex;" /></span></dd></dl></dd></dl></li></ul></dd><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;">Donc, dans tous les cas, <span class="mwe-math-element"><img alt="{\displaystyle F_{i_{s}}\leq n<F_{i_{s}}+F_{i_{s}-1}=F_{i_{s}+1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5de18520546a077cf1d1bfad61c7135b871017d7" style="border: 0px; height: 2.676ex; vertical-align: -0.838ex; width: 30.07ex;" /></span></dd></dl></div><div style="clear: both;"></div></div><div class="NavFrame demonstration" style="background-color: white; border-collapse: collapse; border: thin solid rgb(170, 170, 170); color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin: 1em 2em; overflow: hidden; padding: 0px 1em; text-align: justify;"><div class="NavHead" style="background: transparent; font-weight: 700; padding: 0px;"><strong>Preuve de l'unicité</strong></div><div class="NavContent" style="padding-bottom: 0.4em;"><p style="line-height: inherit; margin: 0.5em 0px;">On raisonne par <a href="https://wikimonde.com/article/Raisonnement_par_r%C3%A9currence#R%C3%A9currence_bien_fond%C3%A9e" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Raisonnement par récurrence">récurrence bien fondée</a> sur l'entier naturel <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>. D'après le lemme, dans une décomposition de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>, le plus grand indice <span class="mwe-math-element"><img alt="k" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c3c9a2c7b599b37105512c5d570edc034056dd40" style="border: 0px; height: 2.176ex; margin: 0px; vertical-align: -0.338ex; width: 1.211ex;" /></span> pour lequel <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">k</sub></span> apparaît (si la décomposition est non vide, <abbr class="abbr nowrap" style="border-bottom: 0px; cursor: help; text-decoration-line: none; text-decoration-style: initial; white-space: nowrap;" title="c’est-à-dire">c.-à-d.</abbr> si <span class="mwe-math-element"><img alt="N\neq 0" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9176ba0573deaebe5dbe9a80523f9bc99af9f61e" style="border: 0px; height: 2.676ex; margin: 0px; vertical-align: -0.838ex; width: 6.325ex;" /></span>) est entièrement déterminé par <span class="mwe-math-element"><img alt="{\displaystyle F_{k}\leq N<F_{k+1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/02772c4dae0e8954ee8e5cd9ca0ff735355df419" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 15.528ex;" /></span>. Par hypothèse de récurrence, la décomposition de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N - F<sub style="line-height: 1;">k</sub></span> (donc le reste de la décomposition de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span>) est également unique.</p></div><div style="clear: both;"></div></div><h3 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: Georgia, Times, serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span id="Repr.C3.A9sentation_des_premiers_entiers"></span><span class="mw-headline" id="Représentation_des_premiers_entiers">Représentation des premiers entiers</span></h3><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Dans la table, <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>R</i>(<i>N</i>)</span> dénote la représentation de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> sous forme de mot binaire.</p><table class="wikitable centre alternance" style="background-color: #f8f9fa; border-collapse: collapse; border: 1px solid rgb(162, 169, 177); color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin: 1em auto;"><tbody><tr style="background-color: #eeeeee;"><th scope="col" style="background-color: #e6e6e6; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span></th><th scope="col" style="background-color: #e6e6e6; border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em; text-align: center;"><span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>R</i>(<i>N</i>)</span></th></tr><tr style="background-color: #fcfcfc;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">0</td></tr><tr style="background-color: #eeeeee;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1</td></tr><tr style="background-color: #fcfcfc;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">2</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10</td></tr><tr style="background-color: #eeeeee;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">3</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">100</td></tr><tr style="background-color: #fcfcfc;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">4</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">101</td></tr><tr style="background-color: #eeeeee;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">5</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1000</td></tr><tr style="background-color: #fcfcfc;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">6</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1001</td></tr><tr style="background-color: #eeeeee;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">7</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">1010</td></tr><tr style="background-color: #fcfcfc;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">8</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10000</td></tr><tr style="background-color: #eeeeee;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">9</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10001</td></tr><tr style="background-color: #fcfcfc;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10010</td></tr><tr style="background-color: #eeeeee;"><td style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">11</td><td align="right" style="border: 1px solid rgb(162, 169, 177); padding: 0.2em 0.4em;">10100</td></tr></tbody></table><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">L'alternance des 0 et 1 dans chacune des colonnes correspond à l'absence ou la présence d'un rectangle dans la figure en tête de la page. La suite des derniers chiffres est</p><center style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px;"><span class="mwe-math-element"><img alt="{\displaystyle 010010100100\cdots }" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f9f4a9d884eacfa045164f0f902a25987570e0d7" style="border: 0px; height: 2.176ex; vertical-align: -0.338ex; width: 17.06ex;" /></span></center><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">C'est le début du <a href="https://wikimonde.com/article/Mot_de_Fibonacci" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Mot de Fibonacci">mot de Fibonacci</a>. En effet, le <i>n</i>-ième symbole du mot de Fibonacci est 0 ou 1 selon que <i>n</i> est « Fibonacci pair » ou « Fibonacci impair ».</p><h2 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px solid rgb(162, 169, 177); font-family: Georgia, Times, serif; font-weight: 400; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span class="mw-headline" id="Variations">Variations</span></h2><h3 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: Georgia, Times, serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span id="Repr.C3.A9sentation_par_des_nombres_de_Fibonacci_d.27indices_n.C3.A9gatifs"></span><span class="mw-headline" id="Représentation_par_des_nombres_de_Fibonacci_d'indices_négatifs">Représentation par des nombres de Fibonacci d'indices négatifs</span></h3><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">La suite des nombres de Fibonacci peut être étendue aux indices négatifs, puisque la relation</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle F_{n}=F_{n-1}+F_{n-2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fa6d281e7a54e08aeffeef7458ddc0884333686" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 18.279ex;" /></span></dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">permet de calculer <span class="mwe-math-element"><img alt="F_{{n-2}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff0954dd46f66d84e032b21cfae83fef10c5fcbb" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 4.814ex;" /></span> à partir de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">n</sub></span> et de <span class="mwe-math-element"><img alt="F_{{n-1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61373b860d2d2e4842b10ac0b1c3f90362c2c7d0" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 4.814ex;" /></span>. On a (voir la <a href="https://wikimonde.com/article/Suite_de_Fibonacci#La_suite_pour_les_nombres_n%C3%A9gatifs" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Suite de Fibonacci">section correspondante de l'article sur les nombres de Fibonacci</a>) :</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle F_{-n}=(-1)^{n+1}F_{n}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28100fff6850e5b4934f783e3b5ec3ece74c55b3" style="border: 0px; height: 3.176ex; vertical-align: -0.838ex; width: 18.549ex;" /></span></dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">La suite complète est <span class="mwe-math-element"><img alt="{\displaystyle \ldots ,-8,5,-3,2,-1,1,0,1,1,2,3,5,8,\ldots }" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6d864a41931f3b5bdcd291df15209d4dfa04e7f3" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 40.845ex;" /></span> <a href="https://wikimonde.com/article/Donald_Knuth" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Donald Knuth">Donald Knuth</a> a remarqué<span class="reference" id="cite_ref-5" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-5" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">5</a></span> que tout <a href="https://wikimonde.com/article/Entier_relatif" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Entier relatif">entier relatif</a> est somme de nombres de Fibonacci d'indices strictement négatifs qu'il appelle « Negafibonacci », la représentation étant unique si deux nombres utilisés ne sont pas consécutifs. Par exemple :</p><ul style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; list-style-image: url("data:image/svg+xml;base64,PHN2ZyB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciIHdpZHRoPSI1IiBoZWlnaHQ9IjEzIj4gPGNpcmNsZSBjeD0iMi41IiBjeT0iOS41IiByPSIyLjUiIGZpbGw9IiMwMDUyOGMiLz4gPC9zdmc+"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin-bottom: 0.1em;"><span class="mwe-math-element"><img alt="{\displaystyle -11=F_{-4}+F_{-6}=(-3)+(-8)}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3805ed7cdab99aeac478dca3d2a0028871644eb" style="border: 0px; height: 2.843ex; vertical-align: -0.838ex; width: 33.225ex;" /></span> ;</li><li style="margin-bottom: 0.1em;"><span class="mwe-math-element"><img alt="{\displaystyle 12=F_{-2}+F_{-7}=(-1)+13}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8faaf3c6edcdf20fc7487a6e86686f6f987c43e8" style="border: 0px; height: 2.843ex; vertical-align: -0.838ex; width: 28.962ex;" /></span> ;</li><li style="margin-bottom: 0.1em;"><span class="mwe-math-element"><img alt="{\displaystyle 24=F_{-1}+F_{-4}+F_{-6}+F_{-9}=1+(-3)+(-8)+34}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65ecb99ab56def7c12cc2628435a212a03d9f3ea" style="border: 0px; height: 2.843ex; vertical-align: -0.838ex; width: 53.921ex;" /></span> ;</li><li style="margin-bottom: 0.1em;"><span class="mwe-math-element"><img alt="{\displaystyle -43=F_{-2}+F_{-7}+F_{-10}=(-1)+13+(-55)}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2542750db7de2eb382aad02adeaf3ad203de00" style="border: 0px; height: 2.843ex; vertical-align: -0.838ex; width: 47.043ex;" /></span>.</li></ul><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Comme plus haut, on associe à la représentation d'un entier <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> un <a href="https://wikimonde.com/article/Mot_%28math%C3%A9matiques%29" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Mot (mathématiques)">mot</a> binaire, dont la <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span>-ième lettre est <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">1</span> si <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">F<sub style="line-height: 1;">n</sub></span> figure dans la représentation de <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> et <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">0</span> sinon. Ainsi, <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">24</span> est représenté par le mot <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">100101001</span>. On observe que l'entier <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> est positif si et seulement si la longueur du mot associé est impaire.</p><h3 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: Georgia, Times, serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span class="mw-headline" id="Multiplication_de_Fibonacci">Multiplication de Fibonacci</span></h3><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;"><a href="https://wikimonde.com/article/Donald_Knuth" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Donald Knuth">Donald Knuth</a><span class="reference" id="cite_ref-6" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-6" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">6</a></span> considère une opération de multiplication <span class="mwe-math-element"><img alt="{\displaystyle a\circ b}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6f270b24a930a6546b42f355ad905d2d7a26d4b3" style="border: 0px; height: 2.176ex; margin: 0px; vertical-align: -0.338ex; width: 4.422ex;" /></span> d'entiers naturels <span class="mwe-math-element"><img alt="a" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" style="border: 0px; height: 1.676ex; margin: 0px; vertical-align: -0.338ex; width: 1.23ex;" /></span> et <span class="mwe-math-element"><img alt="b" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f11423fbb2e967f986e36804a8ae4271734917c3" style="border: 0px; height: 2.176ex; margin: 0px; vertical-align: -0.338ex; width: 0.998ex;" /></span> définie comme suit : étant donné les représentations <span class="mwe-math-element"><img alt="{\displaystyle a=\sum _{i=0}^{k}F_{c_{i}}\;(c_{i}\geq 2)}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/242c6ed5b6786dc3f5e4916dc9631bd2ec551b43" style="border: 0px; height: 7.343ex; margin: 0px; vertical-align: -3.005ex; width: 19.656ex;" /></span> et <span class="mwe-math-element"><img alt="{\displaystyle b=\sum _{j=0}^{l}F_{d_{j}}\;(d_{j}\geq 2)}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5af17e5aebf37f2b738892090fa38e337af0151c" style="border: 0px; height: 7.676ex; margin: 0px; vertical-align: -3.338ex; width: 19.968ex;" /></span> le <i>produit de Fibonacci</i> est l'entier <span class="mwe-math-element"><img alt="{\displaystyle a\circ b=\sum _{i=0}^{k}\sum _{j=0}^{l}F_{c_{i}+d_{j}}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c630cb5b721efccfa8397150e472e42c7298c190" style="border: 0px; height: 7.676ex; margin: 0px; vertical-align: -3.338ex; width: 20.916ex;" /></span>.</p><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Par exemple, comme <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">2 = <i>F</i><sub style="line-height: 1;">3</sub></span> et <span class="texhtml" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">4 = <i>F</i><sub style="line-height: 1;">4</sub> + <i>F</i><sub style="line-height: 1;">2</sub></span>, on a <span class="mwe-math-element"><img alt="{\displaystyle 2\circ 4=F_{3+4}+F_{3+2}=13+5=18}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c9551e55556bb9be626adb67a864cb25bd678c6e" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 34.607ex;" /></span>.</p><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Knuth a prouvé le fait surprenant que cette opération est <a href="https://wikimonde.com/article/Associativit%C3%A9" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Associativité">associative</a>.</p><h3 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: Georgia, Times, serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span class="mw-headline" id="Autres_suites">Autres suites</span></h3><div class="pub_headers_half" style="background-color: white; clear: right; color: #222222; float: right; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; height: 250px; margin: 1px 0px 2px 6px; width: 300px;"><div class="pub_infobox_small tm_pub ban_pub matomoTrackContent" data-content-name="headers half" data-content-piece="PAVE BAS" style="clear: right; float: right; height: 250px; width: 300px;"><div class="tm_bloc" id="90957-19"><div id="sas_26711" style="height: 250px; margin: auto; width: 300px;"><iframe allow="autoplay;fullscreen;" frameborder="0" height="250" id="sas_26711_iframe" marginheight="0" marginwidth="0" scrolling="no" src="about:blank" style="height: 250px; width: 300px;" width="300"></iframe></div></div></div></div><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Zeckendorf prouve l'existence et l'unicité, sous condition, pour la représentation par les <a href="https://wikimonde.com/article/Suite_de_Lucas" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Suite de Lucas">nombres de Lucas</a><span class="reference" id="cite_ref-Zeck_1-2" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-Zeck-1" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">1</a></span>.</p><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">Knuth mentionne que le théorème de Zeckendorf reste vrai pour les <a href="https://wikimonde.com/article/Suite_de_Fibonacci#Suites_de_Fibonacci_g%C3%A9n%C3%A9ralis%C3%A9es" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Suite de Fibonacci">suites de k-bonacci</a>, sous réserve que l'on n'utilise pas <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">k</span> nombres consécutifs d'une telle suite<span class="reference" id="cite_ref-7" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-7" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">7</a></span>.</p><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;"><a href="https://wikimonde.com/article/Aviezri_Fraenkel" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Aviezri Fraenkel">Aviezri Fraenkel</a> a donné un énoncé général qui étend les théorèmes précédents<span class="reference" id="cite_ref-Fraenkel85_8-0" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://wikimonde.com/article/Th%C3%A9or%C3%A8me_de_Zeckendorf#cite_note-Fraenkel85-8" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;">8</a></span> : Soit <span class="mwe-math-element"><img alt="{\displaystyle 1=u_{0}<u_{1}<u_{2}<\cdots }" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/17b28c9e685d3c3dc5e25439fa6357c8c913ca3d" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 23.431ex;" /></span> une <a href="https://wikimonde.com/article/Suite_d%27entiers" style="background: 0px 0px; color: #0645ad; text-decoration-line: none;" title="Suite d'entiers">suite d'entiers</a>. Tout entier naturel <span class="texhtml mvar" style="font-family: "nimbus roman no9 l", "times new roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">N</span> a exactement une représentation de la forme</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle N=d_{k}u_{k}+d_{k-1}u_{k-1}+\cdots +d_{1}u_{1}+d_{0}u_{0}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2fce0fe756c850b09026d23e47bda50e7fce7876" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 42.174ex;" /></span>,</dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">où <span class="mwe-math-element"><img alt="{\displaystyle d_{k},\ldots ,d_{0}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f2375870c874e19e13e2e5085a949662a63d1da8" style="border: 0px; height: 2.509ex; margin: 0px; vertical-align: -0.671ex; width: 9.739ex;" /></span> sont des entiers naturels, pourvu que</p><dl style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><img alt="{\displaystyle d_{i}u_{i}+d_{i-1}u_{i-1}+\cdots +d_{0}u_{0}<u_{i+1}}" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1b5c9f2cd15c98fd799f58596c028793de6373de" style="border: 0px; height: 2.509ex; vertical-align: -0.671ex; width: 35.697ex;" /></span></dd></dl><p style="background-color: white; color: #222222; font-family: "Helvetica Neue", Helvetica, Arial, sans-serif; font-size: 14px; line-height: inherit; margin: 0.5em 0px;">pour <span class="mwe-math-element"><img alt="i\geq 0" aria-hidden="true" class="mwe-math-fallback-image-inline" loading="lazy" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/405e1424cb9c4fc171c433a8e8f04b3e5938e366" style="border: 0px; height: 2.343ex; margin: 0px; vertical-align: -0.505ex; width: 5.063ex;" /></span>.</p><h3 style="background: 0px 0px rgb(255, 255, 255); border-bottom: 1px dotted rgb(170, 170, 170); font-family: Georgia, Times, serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em;"><span id="Syst.C3.A8me_d.27Ostrowski"></span><span class="mw-headline" id="Système_d'Ostrowski">Système d'Ostrowski</span></h3>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-24867038264566836072023-04-23T15:58:00.003+02:002023-04-24T10:12:51.131+02:00CKPLAN<p><span style="font-size: x-large;"> <b>5.55382562855700000×10^(-17)×2^64 = 1024.5</b></span></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0XZ8sy5GyOtfpnun5oXtnoSbOIPmOvtyYCv29fX8EwqLPvctbpId7DZT5DYU6LlCjfjMQGfA9TruFQ9uU_Vbc9CSrk7iTdKjfopAI1EPGgGuQaSVvkytIlNr-05MJjniqS5vS52wUZQKCxJcfCKvm2ae1dB378gL29kzQ7mpBR1_vhUNtZT5JFtCv/s545/64.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="381" data-original-width="545" height="357" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEg0XZ8sy5GyOtfpnun5oXtnoSbOIPmOvtyYCv29fX8EwqLPvctbpId7DZT5DYU6LlCjfjMQGfA9TruFQ9uU_Vbc9CSrk7iTdKjfopAI1EPGgGuQaSVvkytIlNr-05MJjniqS5vS52wUZQKCxJcfCKvm2ae1dB378gL29kzQ7mpBR1_vhUNtZT5JFtCv/w509-h357/64.jpg" width="509" /></a></div><br /><p>2^64=18446744073709551616</p><p> 140737488355328 = 2^47</p><p>762939453125=5^17</p><p><b><span style="font-size: x-large;">1024.50000000000065242385835098112</span></b></p><p><br /></p><p><br /></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-51394572702614706752023-04-21T16:50:00.000+02:002023-04-21T16:50:17.863+02:00Théorème de Dirichlet (séries de Fourier)<p></p><div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen="" class="BLOG_video_class" height="266" src="https://www.youtube.com/embed/EK32jo7i5LQ" width="320" youtube-src-id="EK32jo7i5LQ"></iframe></div><br /> <p></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-81476273577550692732023-03-04T08:01:00.005+01:002023-03-04T08:01:24.089+01:00Triangle d'or<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaXglb74fa2P44dIKXzh6pw3IThdpoZ5Zf821arLiFEPHWkPta-6RVFvlyDVt4XSb49VtCu4nwetWblI4y73nj2EXltBDY9g_TiM2cAD1RRA_Kr56-l9NbJ-D9G0llmI7C8qCdAMPXorndEXnLvRRdNbNI1iEoJOmMugZAFwvTsQWqGAKNlTK2hwWR/s358/Golden_Triangle.svg.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="358" data-original-width="220" height="320" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgaXglb74fa2P44dIKXzh6pw3IThdpoZ5Zf821arLiFEPHWkPta-6RVFvlyDVt4XSb49VtCu4nwetWblI4y73nj2EXltBDY9g_TiM2cAD1RRA_Kr56-l9NbJ-D9G0llmI7C8qCdAMPXorndEXnLvRRdNbNI1iEoJOmMugZAFwvTsQWqGAKNlTK2hwWR/s320/Golden_Triangle.svg.png" width="197" /></a></div><div class="separator" style="clear: both; text-align: center;"><span style="background-color: #f8f9fa; color: #202122; font-family: sans-serif; font-size: 12.3704px; text-align: left;">Triangle d'or. Rapport a/b = nombre d'or φ. Angle au sommet : </span><span class="mwe-math-element" style="background-color: #f8f9fa; color: #202122; font-family: sans-serif; font-size: 12.3704px; text-align: left;"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \theta }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi></mstyle></mrow></semantics></math></span><img alt="\theta " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6e5ab2664b422d53eb0c7df3b87e1360d75ad9af" style="border: 0px; display: inline-block; height: 2.176ex; vertical-align: -0.338ex; width: 1.09ex;" /></span><span style="background-color: #f8f9fa; color: #202122; font-family: sans-serif; font-size: 12.3704px; text-align: left;"> = 36°. Angles de base : 72° chacun.</span></div><div class="separator" style="clear: both; text-align: center;"><span style="background-color: #f8f9fa; color: #202122; font-family: sans-serif; font-size: 12.3704px; text-align: left;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><span style="background-color: #f8f9fa; color: #202122; font-family: sans-serif; font-size: 12.3704px; text-align: left;"><br /></span></div><p></p><p style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin: 0.5em 0px;">Un <b>triangle d'or</b> (aigu) ou <b>triangle sublime</b><span class="reference" id="cite_ref-elam_1-0" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Triangle_d%27or_(g%C3%A9om%C3%A9trie)#cite_note-elam-1" style="background: none; color: #3366cc; text-decoration-line: none;">1</a></span> est un <a href="https://fr.wikipedia.org/wiki/Triangle_isoc%C3%A8le" style="background: none; color: #3366cc; text-decoration-line: none;" title="Triangle isocèle">triangle isocèle</a> dans lequel le rapport de la longueur du côté double à la longueur du côté-base est le <a href="https://fr.wikipedia.org/wiki/Nombre_d%27or" style="background: none; color: #3366cc; text-decoration-line: none;" title="Nombre d'or">nombre d'or</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \varphi }" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi></mstyle></mrow></semantics></math></span><img alt="\varphi " aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/33ee699558d09cf9d653f6351f9fda0b2f4aaa3e" style="border: 0px; display: inline-block; height: 2.176ex; margin: 0px; vertical-align: -0.838ex; width: 1.52ex;" /></span> :</p><div class="separator" style="clear: both; text-align: center;"><dl style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; margin-bottom: 0.5em; margin-top: 0.2em; text-align: start;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle {a \over b}=\varphi ={{1+{\sqrt {5}}} \over 2}\approx 1,618~034}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>�</mi><mi>�</mi></mfrac></mrow><mo>=</mo><mi>�</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow class="MJX-TeXAtom-ORD"><mn>1</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msqrt><mn>5</mn></msqrt></mrow></mrow><mn>2</mn></mfrac></mrow><mo>≈</mo><mn>1</mn><mo>,</mo><mn>618</mn><mtext> </mtext><mn>034</mn></mstyle></mrow></semantics></math></span><img alt="{\displaystyle {a \over b}=\varphi ={{1+{\sqrt {5}}} \over 2}\approx 1,618~034}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6fbdd6dcc5d782cf7fff5c52b6e67c19926a0a71" style="border: 0px; display: inline-block; height: 6.009ex; vertical-align: -2.005ex; width: 30.571ex;" /></span></dd></dl></div><p><br /></p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZzraPfiazjGlx0hBH0QX9AYV7rcgMNPYYU2BJ7FziUtJn6fwlBJCFyF_c6KxPPKkSnJJmOucUo5C4IJgo0KAS-BakNE1eP7ap6yaFRkw-p-zB7FkEq5GMjEJ_-G2hwBsETcAAxqRdWASLFec0El7vMv_noYlKRna7C5zOp4akweLvgZj_kQjDhhHe/s220/Pavage_d'un_pentagone_r%C3%A9gulier_par_3_triangles.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="220" data-original-width="220" height="220" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZzraPfiazjGlx0hBH0QX9AYV7rcgMNPYYU2BJ7FziUtJn6fwlBJCFyF_c6KxPPKkSnJJmOucUo5C4IJgo0KAS-BakNE1eP7ap6yaFRkw-p-zB7FkEq5GMjEJ_-G2hwBsETcAAxqRdWASLFec0El7vMv_noYlKRna7C5zOp4akweLvgZj_kQjDhhHe/s1600/Pavage_d'un_pentagone_r%C3%A9gulier_par_3_triangles.png" width="220" /></a></div><br /><p></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-66885665183479969672023-02-27T08:24:00.005+01:002022-10-02T08:11:41.861+02:0010! et temps Numération factorielle<div class="separator" style="clear: both; text-align: center;">
<a href="https://1.bp.blogspot.com/-M6vADPf_eYg/Xli8l1wcIqI/AAAAAAABMUw/3LHu54qHgWkQIaagugHYmDW1YRQD2Pr_QCLcBGAsYHQ/s1600/fact.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="89" data-original-width="333" height="170" src="https://1.bp.blogspot.com/-M6vADPf_eYg/Xli8l1wcIqI/AAAAAAABMUw/3LHu54qHgWkQIaagugHYmDW1YRQD2Pr_QCLcBGAsYHQ/s640/fact.jpg" width="640" /></a></div>
<span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Système de numération factorielle </span><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Article principal: </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Système de numération factorielle</span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"> Une autre proposition est le soi-disant système numérique factoriel : </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Base 8 7 6 5 4 3 2 1 </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Valeur de position 7! 6! 5! 4! 3! 2! 1! 0!</span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"> Placez la valeur en décimal 5040 720 120 24 6 2 1 1 </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Chiffre le plus élevé autorisé 7 6 5 4 3 2 1 0 </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Par exemple, le plus grand nombre qui pourrait être représenté avec six chiffres serait 543210,</span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"> ce qui équivaut à 719 en décimal : 5 × 5! + 4 × 4! + 3 × 3! + 2 × 2! + 1 × 1! </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"><br /></span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Ce n'est peut-être pas clair à première vue, mais le système de numérotation factorielle est sans ambiguïté et complet. </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"><br /></span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Chaque nombre peut être représenté d'une et une seule manière car la somme des factorielles respectives multipliée par l'indice est toujours la prochaine factorielle moins un:</span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"> Il existe une correspondance naturelle entre les entiers 0, ..., n ! - 1 et permutations de n éléments dans l'ordre lexicographique, qui utilise la représentation factorielle de l'entier, suivie d'une interprétation sous forme de code de Lehmer .</span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"><br /></span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"> L'équation ci-dessus est un cas particulier de la règle générale suivante pour toute représentation de base de base (standard ou mixte) qui exprime le fait que toute représentation de base de base (standard ou mixte) est sans ambiguïté et complète. </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"><br /></span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Chaque nombre peut être représenté d'une et une seule manière car la somme des poids respectifs multipliée par l'indice est toujours le poids suivant moins un: , où , qui peut être facilement prouvée par induction mathématique . </span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;"><br /></span></div><div><span face="sans-serif" style="background-color: #f6f6f6; font-size: 16px;">Base mixte - https://fr.other.wiki/wiki/Mixed_Radix#Primorial_number_system</span><br />
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6!=720<br />
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3628800=10!=2^8*3^4*5^2*7^1<br />
10! secondes = 6 semaines=42 jours; 1 semaine = 7 jours,1 jour= 24 heures ....<br />
12 mois de 30 jours environ ! par an,2 semestres 4 trimestres .......<br />
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10!=6*7 jours<br />
10! = 6!×7!<br />
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<b>Et 30 solutions entières à (10!/y)^1/2</b><br />
<b>par exemple (10!/567)^1/2=80</b><br />
<b>Idem pour (11!/77)^1/2 ; 30 solutions entières</b><br />
<b>par exemple (11!/6237)^1/2=80</b><br />
<b>36 solutions pour (12!/y)^1/2</b><br />
<b>par exemple </b><br />
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les 360°, temps et fréquence , cycles par seconde .....<br />
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Mécanique "classique" : nombre définissant distances etc ..i.e; :.la matière contenant du temps (durée de vie)<br />
mécanique quantique : chiffres définissant le temps, via la factorielle, puis la matière......<br />
Le chiffre n'exprime que du temps le nombre exprime temps et matière et .... position (x,y,z) ou (x,y)<br />
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Une autre manière de créer le nombre : triangle de Pascal et son lien avec Fibonacci et stirling<br />
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Neuf fois sept =63=777 777 777 ( 3^2×7×37×333667)<br />
différent de sept fois neuf = 63 =999 999 9 ( 3^2×239×4649)<br />
suite LS de Conway<br />
<b>la pensée, la parole,l'écrit et le nombre .....</b><br />
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le temps donne le chiffre le nombre donne le matériel .....<br />
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12345679=37*333667 le 8 est absent<br />
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777777777=3*777*333667<br />
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1/81= 0.012345679 012345679 01....<br />
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les sept,7, présents dans la Bible , apocalypse ...<br />
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30*4!*5!= 86400 secondes = 1 jour = ....<br />
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</span></b><dd style="font-family: Georgia, Helvetica, Arial, sans-serif;"><div style="font-variant-east-asian: normal; font-variant-numeric: normal; padding: 0px;">
<span face="sans-serif" style="color: #1b1b21;"><span style="font-size: large;"><b>(10!+1)/7=518400.1428571428571428571428571428571428571428571428571428...</b></span></span></div>
</dd><dd style="font-family: Georgia, Helvetica, Arial, sans-serif;"><span face="sans-serif" style="color: #1b1b21;"><span style="font-size: large;"><b>518400 secondes = 6 jours</b></span></span></dd><dd style="font-family: Georgia, Helvetica, Arial, sans-serif; font-size: 13px;"><span face="sans-serif" style="color: #1b1b21;"><span style="font-size: 13.2px;"><br /></span></span></dd><dd style="font-family: Georgia, Helvetica, Arial, sans-serif; font-size: 13px;"><span face="sans-serif" style="color: #1b1b21;"><span style="font-size: 13.2px;"><br /></span></span></dd><dd style="font-family: Georgia, Helvetica, Arial, sans-serif; font-size: 13px;"><span face="sans-serif" style="color: #1b1b21;"><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-QBABDUsMVnk/YDNv-AR03AI/AAAAAAABPMQ/fq2MknCuHOgh0k3fHcEytl1tr6eHOrdUgCLcBGAsYHQ/s681/142857.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="681" data-original-width="365" height="755" src="https://1.bp.blogspot.com/-QBABDUsMVnk/YDNv-AR03AI/AAAAAAABPMQ/fq2MknCuHOgh0k3fHcEytl1tr6eHOrdUgCLcBGAsYHQ/w406-h755/142857.jpg" width="406" /></a></div><br /><span style="font-size: 13.2px;"><br /></span></span></dd><dd style="font-family: Georgia, Helvetica, Arial, sans-serif; font-size: 13px;"><span face="sans-serif" style="color: #1b1b21;"><span style="font-size: 13.2px;"><br /></span></span></dd><dd style="font-family: Georgia, Helvetica, Arial, sans-serif; font-size: 13px;"><span face="sans-serif" style="color: #1b1b21;"><span style="font-size: 13.2px;"><br /></span></span></dd><br />
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777777777=9*86419753<br />
777777777=3^2×7×37×333667<br />
777777777 divides 100^9 - 1.<br />
factors | 777777777 :<br />
1 | 3 | 7 | 9 | 21 | 37 | 63 | 111 | 259 | 333 | 777 | 2331 | 333667 | 1001001 | 2335669 | 3003003 | 7007007 | 12345679 | 21021021 | 37037037 | 86419753 | 111111111 | 259259259 | 777777777 (24 divisors)<br />
7.7715611723760957829654216766357421875 × 10^-16=7/2^53<div><br /></div><div><br /></div><div><h3 style="background-color: white; color: #0066dd; font-family: Arial, Helvetica, sans-serif; margin: 15px 0px 10px; max-width: 100%; position: relative; text-align: justify;"><span class="mw-headline" id="Th.C3.A9or.C3.A8me_de_Wilson" style="max-width: 100%;">Théorème de Wilson</span></h3><p style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: justify;">Un exemple d'équation diophantienne utilisant ces outils pour sa résolution est le théorème de Wilson. Il correspond à la résolution de l'équation suivante, le signe ! désignant la fonction factorielle :</p><center style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%;"><img alt="(x - 1)! + 1 = y\cdot x\;" class="tex" src="https://www.techno-science.net/illustration/Definition/inconnu/e/e6106784261631ff82bcdc0654de5b85.png" style="background-attachment: initial; background-clip: initial; background-image: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; border: 0px; box-shadow: rgba(0, 0, 0, 0.1) 1px 1px 5px; height: auto; max-width: 100%; padding: 5px; vertical-align: middle;" /></center><p style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: justify;">Les seules valeurs de <i style="max-width: 100%;">x</i> différentes de <i style="max-width: 100%;">un</i> vérifiant cette équation sont les nombres premiers.</p><p style="background-color: white; color: #222222; font-family: Arial, Helvetica, sans-serif; font-size: 16px; max-width: 100%; text-align: justify;"><br /></p><h3 style="background-color: white; border-bottom: 1px dotted rgb(170, 170, 170); color: #222222; font-family: sans-serif; font-size: 1.2em; line-height: 1.6; margin: 0.3em 0px 0px; overflow: hidden; padding-bottom: 0px; padding-top: 0.5em; position: relative;"><span class="mw-headline" id="Exemples">Exemples</span><span class="mw-editsection" style="font-size: small; font-weight: normal; line-height: 1em; margin-left: 1em; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a class="mw-editsection-visualeditor" href="https://fr.wikipedia.org/w/index.php?title=Th%C3%A9or%C3%A8me_de_Wilson&veaction=edit&section=3" style="background: none; color: #0645ad; text-decoration-line: none; white-space: nowrap;" title="Modifier la section : Exemples">modifier</a><span class="mw-editsection-divider" style="color: #54595d;"> | </span><a href="https://fr.wikipedia.org/w/index.php?title=Th%C3%A9or%C3%A8me_de_Wilson&action=edit&section=3" style="background: none; color: #0645ad; text-decoration-line: none; white-space: nowrap;" title="Modifier la section : Exemples">modifier le code</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h3><ul style="background-color: white; color: #202122; font-family: sans-serif; font-size: 14px; line-height: 1.4; list-style-image: url("/w/skins/Vector/resources/skins.vector.styles/images/bullet-icon.svg?d4515"); margin: 0.3em 0px 0px 1.6em; padding: 0px;"><li style="margin: 0px 0px 0.1em; padding: 0px;">Si <i>p</i> est égal à 2, alors (<i>p</i> – 1)! + 1 est égal à 2, un multiple de 2.</li><li style="margin: 0px 0px 0.1em; padding: 0px;">Si <i>p</i> est égal à 3, alors (<i>p</i> – 1)! + 1 est égal à 3, un multiple de 3.</li><li style="margin: 0px 0px 0.1em; padding: 0px;">Si <i>p</i> est égal à 4, alors (<i>p</i> – 1)! + 1 est égal à 7 qui n'est pas multiple de 4.</li><li style="margin: 0px 0px 0.1em; padding: 0px;">Si <i>p</i> est égal à 5, alors (<i>p</i> – 1)! + 1 est égal à 25, un multiple de 5.</li><li style="margin: 0px 0px 0.1em; padding: 0px;">Si <i>p</i> est égal à 6, alors (<i>p</i> – 1)! + 1 est égal à 121 qui n'est pas multiple de 6.</li><li style="margin: 0px 0px 0.1em; padding: 0px;">Si <i>p</i> est égal à 17, alors (<i>p</i> – 1)! + 1 est égal à 20 922 789 888 001, un multiple de 17 car <span class="nowrap" style="white-space: nowrap;">17 × 1 230 752 346 353 = 20 922 789 888 001</span>.</li></ul><div><span style="color: #202122; font-family: sans-serif;"><span style="font-size: 14px;"><br /></span></span></div></div></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;">Le système de nombres factoriels fournit une représentation unique pour chaque nombre naturel, avec la restriction donnée sur les «chiffres» utilisés.</span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;">Aucun nombre ne peut être représenté de plus d'une manière car la somme des factorielles consécutives multipliée par leur indice est toujours la prochaine factorielle moins un: </span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;">Cela peut être facilement prouvé avec une induction mathématique , ou simplement en remarquant que : les termes suivants s'annulent, laissant le premier et le dernier terme (voir Série télescopique ) </span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;">Cependant, lorsque vous utilisez des chiffres arabes pour écrire les chiffres (sans inclure les indices comme dans les exemples ci-dessus), leur simple concaténation devient ambiguë pour les nombres ayant un "chiffre" supérieur à 9. </span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;">Le plus petit exemple est le nombre 10 × 10! = 36,288,000 10 , qui peut s'écrire A0000000000 ! = 10: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0 ! ,</span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;"> mais pas 100000000000 ! = 1: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0: 0 ! ce qui dénote 11! = 39 916 800 10</span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;"><br /></span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;"> . Ainsi, en utilisant les lettres A – Z pour désigner les chiffres 10, 11, 12, ..., 35 comme dans les autres bases-N, le plus grand nombre représentable est 36 × 36! - 1.</span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;"><br /></span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;"> Pour des nombres arbitrairement plus grands, il faut choisir une base pour représenter les chiffres individuels, disons décimal, et fournir une marque de séparation entre eux (par exemple en indiquant chaque chiffre par sa base, également donnée en décimal, comme 2 4 0 3 1 2 0 1 , ce nombre peut également s'écrire 2: 0: 1: 0 ! )</span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;"> En fait, le système numérique factoriel lui-même n'est pas vraiment un système numérique dans le sens de fournir une représentation pour tous les nombres naturels en utilisant seulement un alphabet fini de symboles, car il nécessite une marque de séparation supplémentaire.</span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;">Permutations Il existe une correspondance naturelle entre les entiers 0, ..., n ! - 1 (ou de manière équivalente les nombres à n chiffres en représentation factorielle) et permutations de n éléments dans l' ordre lexicographique , lorsque les entiers sont exprimés sous forme factoradique. </span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;"><br /></span></div><div><span style="background-color: #f6f6f6; font-family: sans-serif; font-size: 16px;">Ce mappage a été appelé le code Lehmer (ou table d'inversion). Système de numération factorielle - https://fr.other.wiki/wiki/Factorial_number_system</span></div>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-29601583042875106552023-02-17T07:49:00.000+01:002023-02-18T07:57:26.087+01:00Charles Hermite<p> <span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">De manière cohérente, </span></p><p><span face="sans-serif" style="background-color: white; color: #202122;"><span style="font-size: 14px;">Hermite voit le</span><b><span style="font-size: medium;"> travail du mathématicien</span></b><span style="font-size: 14px;"> comme proche de celui du naturaliste : </span></span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><br /></span></p><p><span face="sans-serif" style="background-color: white; color: #202122;"><b><span style="font-size: x-large;">récolter des exemples,</span></b></span></p><p><span face="sans-serif" style="background-color: white; color: #202122;"><b><span style="font-size: x-large;"> les comparer et les observer, </span></b></span></p><p><span face="sans-serif" style="background-color: white; color: #202122;"><b><span style="font-size: x-large;">les classer.</span></b></span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> Plusieurs de ses positions sont d'ailleurs partagées par ses correspondants, </span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">comme Thomas Stieltjes ou </span><a href="https://fr.wikipedia.org/wiki/Leo_K%C3%B6nigsberger" style="background: none rgb(255, 255, 255); color: #0645ad; font-family: sans-serif; font-size: 14px; text-decoration-line: none;" title="Leo Königsberger">Leo Königsberger</a><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;">.</span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><br /></span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"> Hermite approuve par exemple une phrase de Königsberger : </span></p><p><span face="sans-serif" style="background-color: white; color: #202122; font-size: 14px;"><br /></span></p><p><b><span style="font-size: large;"><span face="sans-serif" style="background-color: white; color: #202122;">« Il me semble que la tâche principale, actuellement, de même que pour l’histoire naturelle descriptive, consiste à amasser le plus possible de matériaux, et à découvrir des principes en classant et décrivant ces matériaux</span><span class="reference" face="sans-serif" id="cite_ref-16" style="background-color: white; color: #202122; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Charles_Hermite#cite_note-16" style="background: none; color: #0645ad; text-decoration-line: none;">16</a></span><span face="sans-serif" style="background-color: white; color: #202122;"> ».</span></span></b></p>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-91734453326668865882023-02-07T15:49:00.003+01:002023-02-07T15:49:25.813+01:00Nombre pentagonal<p> </p><header class="mw-body-header vector-page-titlebar" style="align-items: center; background-color: white; box-shadow: rgb(162, 169, 177) 0px 1px; color: #202122; display: flex; font-family: sans-serif; font-size: 16px; justify-content: flex-end;"><h1 class="firstHeading mw-first-heading" id="firstHeading" style="border: 0px; color: black; flex-grow: 1; font-family: "Linux Libertine", Georgia, Times, serif; font-size: 1.8em; font-weight: normal; line-height: 1.3; margin: 0px; overflow-wrap: break-word; overflow: hidden; padding: 0px;"><span class="mw-page-title-main">Nombre pentagonal</span></h1><div class="vector-menu vector-dropdown vector-menu-dropdown mw-portlet mw-portlet-lang mw-ui-icon-flush-right" id="p-lang-btn" style="box-sizing: border-box; flex-shrink: 0; float: right; height: 2em; margin-right: -12px; margin-top: 2px; position: relative;"><input aria-haspopup="true" aria-label="Aller à un article dans une autre langue. 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font-size: 12.3704px; line-height: 1.4em; padding: 3px; text-align: left;"><div class="magnify" style="float: right; margin-left: 3px; margin-right: 0px;"><a class="internal" href="https://fr.wikipedia.org/wiki/Fichier:Nombre_pentagon.svg" style="background: url("/w/resources/src/mediawiki.skinning/images/magnify-clip-ltr.svg?8330e"); color: #3366cc; display: block; height: 11px; overflow: hidden; text-decoration-line: none; text-indent: 15px; user-select: none; white-space: nowrap; width: 15px;" title="Agrandir"></a></div>Représentation des quatre premiers nombres pentagonaux : la représentation du <i>n</i>-ième s'obtient en entourant la précédente d'un pentagone comportant 3<i>n</i> – 2 nouveaux points.</div></div></div><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin: 0.5em 0px 1.3em 1.4em; width: auto;"><div class="thumbinner" style="background-color: #f8f9fa; border: 1px solid rgb(200, 204, 209); font-size: 13.16px; min-width: 100px; overflow: hidden; padding: 3px; text-align: center; width: 222px;"><a class="image" href="https://commons.wikimedia.org/wiki/File:Pentagonal_number_22_as_sum_of_gnomons.svg?uselang=fr" style="background: none; color: #3366cc; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="189" data-file-width="200" decoding="async" height="208" src="https://upload.wikimedia.org/wikipedia/commons/thumb/2/28/Pentagonal_number_22_as_sum_of_gnomons.svg/220px-Pentagonal_number_22_as_sum_of_gnomons.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Pentagonal_number_22_as_sum_of_gnomons.svg/330px-Pentagonal_number_22_as_sum_of_gnomons.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Pentagonal_number_22_as_sum_of_gnomons.svg/440px-Pentagonal_number_22_as_sum_of_gnomons.svg.png 2x" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="220" /></a><div class="thumbcaption" style="border: 0px; font-size: 12.3704px; line-height: 1.4em; padding: 3px; text-align: left;"><div class="magnify" style="float: right; margin-left: 3px; margin-right: 0px;"><a class="internal" href="https://fr.wikipedia.org/wiki/Fichier:Pentagonal_number_22_as_sum_of_gnomons.svg" style="background: url("/w/resources/src/mediawiki.skinning/images/magnify-clip-ltr.svg?8330e"); color: #3366cc; display: block; height: 11px; overflow: hidden; text-decoration-line: none; text-indent: 15px; user-select: none; white-space: nowrap; width: 15px;" title="Agrandir"></a></div>Les quatre premiers nombres pentagonaux sont<center><span style="color: red;">1</span>, 1 + <span style="color: orange;">4</span> = 5, 5 + <span style="color: green;">7</span> = 12 et 12 + <span style="color: blue;">10</span> = 22.</center></div></div></div><p style="margin: 0.5em 0px;">En <a href="https://fr.wikipedia.org/wiki/Math%C3%A9matiques" style="background: none; color: #3366cc; text-decoration-line: none;" title="Mathématiques">mathématiques</a>, un <b>nombre pentagonal</b> est un <a href="https://fr.wikipedia.org/wiki/Nombre_figur%C3%A9" style="background: none; color: #3366cc; text-decoration-line: none;" title="Nombre figuré">nombre figuré</a> qui peut être représenté par un <a class="mw-redirect" href="https://fr.wikipedia.org/wiki/Pentagone_(figure)" style="background: none; color: #3366cc; text-decoration-line: none;" title="Pentagone (figure)">pentagone</a>. Pour tout <a href="https://fr.wikipedia.org/wiki/Entier_naturel" style="background: none; color: #3366cc; text-decoration-line: none;" title="Entier naturel">entier</a> <i>n</i> ≥ 1, d'après les <a href="https://fr.wikipedia.org/wiki/Nombre_polygonal" style="background: none; color: #3366cc; text-decoration-line: none;" title="Nombre polygonal">formules générales pour les nombres polygonaux</a>, le <i>n</i>-ième nombre pentagonal est donc la <a href="https://fr.wikipedia.org/wiki/Suite_arithm%C3%A9tique#Somme_des_termes" style="background: none; color: #3366cc; text-decoration-line: none;" title="Suite arithmétique">somme des <i>n</i> premiers termes de la suite arithmétique</a> de premier terme 1 et de raison 3<span class="reference" id="cite_ref-1" style="font-size: 0.8em; line-height: 1; padding-left: 1px; position: relative; top: -5px; unicode-bidi: isolate; vertical-align: text-top; white-space: nowrap;"><a href="https://fr.wikipedia.org/wiki/Nombre_pentagonal#cite_note-1" style="background: none; color: #3366cc; text-decoration-line: none;">1</a></span> :</p><center><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle P_{5,n}=1+4+\dots +(3n-2)={n(3n-1) \over 2}={\frac {1}{3}}~P_{3,3n-1},}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>5</mn><mo>,</mo><mi>�</mi></mrow></msub><mo>=</mo><mn>1</mn><mo>+</mo><mn>4</mn><mo>+</mo><mo>⋯</mo><mo>+</mo><mo stretchy="false">(</mo><mn>3</mn><mi>�</mi><mo>−</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>�</mi><mo stretchy="false">(</mo><mn>3</mn><mi>�</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><mn>2</mn></mfrac></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>1</mn><mn>3</mn></mfrac></mrow><mtext> </mtext><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn><mo>,</mo><mn>3</mn><mi>�</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle P_{5,n}=1+4+\dots +(3n-2)={n(3n-1) \over 2}={\frac {1}{3}}~P_{3,3n-1},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/39191fe296c820c1cae02d304a35e9273f3e8b87" style="border: 0px; display: inline-block; height: 5.676ex; vertical-align: -1.838ex; width: 55.963ex;" /></span></center><p style="margin: 0.5em 0px;">soit le tiers du (3<i>n</i> – 1)-ième <a href="https://fr.wikipedia.org/wiki/Nombre_triangulaire" style="background: none; color: #3366cc; text-decoration-line: none;" title="Nombre triangulaire">nombre triangulaire</a> et les dix premiers sont <a href="https://fr.wikipedia.org/wiki/1_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="1 (nombre)">1</a>, <a href="https://fr.wikipedia.org/wiki/5_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="5 (nombre)">5</a>, <a href="https://fr.wikipedia.org/wiki/12_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="12 (nombre)">12</a>, <a href="https://fr.wikipedia.org/wiki/22_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="22 (nombre)">22</a>, <a href="https://fr.wikipedia.org/wiki/35_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="35 (nombre)">35</a>, <a href="https://fr.wikipedia.org/wiki/51_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="51 (nombre)">51</a>, <a href="https://fr.wikipedia.org/wiki/70_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="70 (nombre)">70</a>, <a href="https://fr.wikipedia.org/wiki/92_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="92 (nombre)">92</a>, <a href="https://fr.wikipedia.org/wiki/117_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="117 (nombre)">117</a> et <a href="https://fr.wikipedia.org/wiki/145_(nombre)" style="background: none; color: #3366cc; text-decoration-line: none;" title="145 (nombre)">145</a> (suite <a class="extiw" href="https://oeis.org/A000326" style="background: none; color: #3366cc; text-decoration-line: none;" title="oeis:A000326">A000326</a> de l'<a href="https://fr.wikipedia.org/wiki/Encyclop%C3%A9die_en_ligne_des_suites_de_nombres_entiers" style="background: none; color: #3366cc; text-decoration-line: none;" title="Encyclopédie en ligne des suites de nombres entiers">OEIS</a>).</p><p style="margin: 0.5em 0px;">Les nombres pentagonaux sont importants dans la théorie des <a href="https://fr.wikipedia.org/wiki/Partition_d%27un_entier" style="background: none; color: #3366cc; text-decoration-line: none;" title="Partition d'un entier">partitions d'entiers</a> d'<a href="https://fr.wikipedia.org/wiki/Leonhard_Euler" style="background: none; color: #3366cc; text-decoration-line: none;" title="Leonhard Euler">Euler</a> et interviennent par exemple dans son <a href="https://fr.wikipedia.org/wiki/Th%C3%A9or%C3%A8me_des_nombres_pentagonaux" style="background: none; color: #3366cc; text-decoration-line: none;" title="Théorème des nombres pentagonaux">théorème des nombres pentagonaux</a>.</p><h2 style="border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span class="mw-headline" id="Test_des_nombres_pentagonaux">Test des nombres pentagonaux</span><span class="mw-editsection" style="font-family: sans-serif; font-size: small; line-height: 0; margin-left: 1em; margin-right: 0px; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a class="mw-editsection-visualeditor" href="https://fr.wikipedia.org/w/index.php?title=Nombre_pentagonal&veaction=edit&section=1" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Modifier la section : Test des nombres pentagonaux">modifier</a><span class="mw-editsection-divider" style="color: #54595d;"> | </span><a href="https://fr.wikipedia.org/w/index.php?title=Nombre_pentagonal&action=edit&section=1" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Modifier la section : Test des nombres pentagonaux">modifier le code</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h2><p style="margin: 0.5em 0px;">Un <a href="https://fr.wikipedia.org/wiki/Nombre_r%C3%A9el" style="background: none; color: #3366cc; text-decoration-line: none;" title="Nombre réel">réel</a> positif <i>x</i> est pentagonal si et seulement si l'<a href="https://fr.wikipedia.org/wiki/%C3%89quation_du_second_degr%C3%A9" style="background: none; color: #3366cc; text-decoration-line: none;" title="Équation du second degré">équation du second degré</a> 3<i>n</i><sup style="line-height: 1;">2</sup> – <i>n</i> – 2<i>x</i> possède une solution entière <i>n</i> > 0, c'est-à-dire si le réel suivant est entier :</p><p style="margin: 0.5em 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle n={\frac {1+{\sqrt {24x+1}}}{6}}.}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mn>1</mn><mo>+</mo><mrow class="MJX-TeXAtom-ORD"><msqrt><mn>24</mn><mi>�</mi><mo>+</mo><mn>1</mn></msqrt></mrow></mrow><mn>6</mn></mfrac></mrow><mo>.</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle n={\frac {1+{\sqrt {24x+1}}}{6}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c800d1ad2e357757e68ee6a1e565a5c513fa557f" style="border: 0px; display: inline-block; height: 5.843ex; margin: 0px; vertical-align: -1.838ex; width: 19.572ex;" /></span></p><p style="margin: 0.5em 0px;">Lorsque <i>n</i> est entier, <i>x</i> est le <i>n</i>-ième nombre pentagonal.</p><h2 style="border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span id="Nombres_pentagonaux_g.C3.A9n.C3.A9ralis.C3.A9s"></span><span class="mw-headline" id="Nombres_pentagonaux_généralisés">Nombres pentagonaux généralisés</span><span class="mw-editsection" style="font-family: sans-serif; font-size: small; line-height: 0; margin-left: 1em; margin-right: 0px; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a class="mw-editsection-visualeditor" href="https://fr.wikipedia.org/w/index.php?title=Nombre_pentagonal&veaction=edit&section=2" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Modifier la section : Nombres pentagonaux généralisés">modifier</a><span class="mw-editsection-divider" style="color: #54595d;"> | </span><a href="https://fr.wikipedia.org/w/index.php?title=Nombre_pentagonal&action=edit&section=2" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Modifier la section : Nombres pentagonaux généralisés">modifier le code</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h2><p style="margin: 0.5em 0px;">Les nombres pentagonaux généralisés sont les nombres de la forme <i>n</i>(3<i>n</i> – 1)/2, mais avec <i>n</i> <a href="https://fr.wikipedia.org/wiki/Entier_relatif" style="background: none; color: #3366cc; text-decoration-line: none;" title="Entier relatif">entier relatif</a>, ou encore : les nombres de la forme <i>n</i>(3<i>n</i> ± 1)/2 avec <i>n</i> entier naturel. Les vingt premiers termes de cette <a href="https://fr.wikipedia.org/wiki/Suite_d%27entiers" style="background: none; color: #3366cc; text-decoration-line: none;" title="Suite d'entiers">suite d'entiers</a> sont 0, <b>1</b>, 2, <b>5</b>, 7, <b>12</b>, 15, <b>22</b>, 26, <b>35</b>, 40, <b>51</b>, 57, <b>70</b>, 77, <b>92</b>, 100, <b>117</b>, 126 et <b>145</b> (suite <a class="extiw" href="https://oeis.org/A001318" style="background: none; color: #3366cc; text-decoration-line: none;" title="oeis:A001318">A001318</a> de l'<a href="https://fr.wikipedia.org/wiki/Encyclop%C3%A9die_en_ligne_des_suites_de_nombres_entiers" style="background: none; color: #3366cc; text-decoration-line: none;" title="Encyclopédie en ligne des suites de nombres entiers">OEIS</a>).</p></div></div></div>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0tag:blogger.com,1999:blog-1891157076635441651.post-48767800367691762442023-02-07T15:47:00.001+01:002023-02-19T07:28:55.701+01:00Nombre pentatopique<p> </p><div class="separator" style="clear: both; text-align: center;"><a href="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMJK4Th2pewqbp1hi2sJucxnsul_Ei0kWnUGPYrByoan3ZKas9n9Aut-uyft9tlxWONMpCc4a5yq1sSPDSuZX_3GybWvlfkjeBiblvJ2Pwdw_3g1ocekGJ9V4rkOW8o5oQEgrfEQRQtTtXmEJ3tsnX6Xg4Rx6xPgh15XICx1o5z7Z2hNg9AEEUavzQ/s501/pentatope.jpg" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="501" data-original-width="292" height="599" src="https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEgMJK4Th2pewqbp1hi2sJucxnsul_Ei0kWnUGPYrByoan3ZKas9n9Aut-uyft9tlxWONMpCc4a5yq1sSPDSuZX_3GybWvlfkjeBiblvJ2Pwdw_3g1ocekGJ9V4rkOW8o5oQEgrfEQRQtTtXmEJ3tsnX6Xg4Rx6xPgh15XICx1o5z7Z2hNg9AEEUavzQ/w350-h599/pentatope.jpg" width="350" /></a></div><p></p><header class="mw-body-header vector-page-titlebar" style="align-items: center; background-color: white; box-shadow: rgb(162, 169, 177) 0px 1px; color: #202122; display: flex; font-family: sans-serif; font-size: 16px; justify-content: flex-end;"><h1 class="firstHeading mw-first-heading" id="firstHeading" style="border: 0px; color: black; flex-grow: 1; font-family: "Linux Libertine", Georgia, Times, serif; font-size: 1.8em; font-weight: normal; line-height: 1.3; margin: 0px; overflow-wrap: break-word; overflow: hidden; padding: 0px;"><span class="mw-page-title-main">Pentatope number</span></h1><div class="vector-menu vector-dropdown vector-menu-dropdown mw-portlet mw-portlet-lang mw-ui-icon-flush-right" id="p-lang-btn" style="box-sizing: border-box; flex-shrink: 0; float: right; height: 2em; margin-right: -12px; margin-top: 2px; position: relative;"><input aria-haspopup="true" aria-label="Go to an article in another language. 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color: #202122; font-family: sans-serif; font-size: calc(0.875em); line-height: 1.6; position: relative; z-index: 0;"><div class="vector-body-before-content"><div class="mw-indicators" style="float: right; font-size: 0.875em; line-height: 1.6; position: relative; z-index: 1;"></div><div class="noprint" id="siteSub" style="font-size: 12.8px; margin-top: 8px;">From Wikipedia, the free encyclopedia</div></div><div id="contentSub" style="color: #54595d; font-size: 11.76px; line-height: 1.2em; margin: 8px 0px 0px; width: auto;"><div id="mw-content-subtitle"></div></div><div class="mw-body-content mw-content-ltr" dir="ltr" id="mw-content-text" lang="en" style="margin-top: 16px;"><div class="mw-parser-output"><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin: 0.5em 0px 1.3em 1.4em; width: auto;"><div class="thumbinner" style="background-color: #f8f9fa; border: 1px solid rgb(200, 204, 209); font-size: 13.16px; min-width: 100px; overflow: hidden; padding: 3px; text-align: center; width: 222px;"><a class="image" href="https://en.wikipedia.org/wiki/File:Pascal_triangle_simplex_numbers.svg" style="background: none; color: #3366cc; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="341" data-file-width="512" decoding="async" height="147" src="https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pascal_triangle_simplex_numbers.svg/220px-Pascal_triangle_simplex_numbers.svg.png" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pascal_triangle_simplex_numbers.svg/330px-Pascal_triangle_simplex_numbers.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Pascal_triangle_simplex_numbers.svg/440px-Pascal_triangle_simplex_numbers.svg.png 2x" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="220" /></a><div class="thumbcaption" style="border: 0px; font-size: 12.3704px; line-height: 1.4em; padding: 3px; text-align: left;"><div class="magnify" style="float: right; margin-left: 3px; margin-right: 0px;"><a class="internal" href="https://en.wikipedia.org/wiki/File:Pascal_triangle_simplex_numbers.svg" style="background-attachment: initial; background-clip: initial; background-color: initial; background-origin: initial; background-position: initial; background-repeat: initial; background-size: initial; background: url("/w/resources/src/mediawiki.skinning/images/magnify-clip-ltr.svg?8330e"); color: #3366cc; display: block; height: 11px; overflow: hidden; text-decoration-line: none; text-indent: 15px; user-select: none; white-space: nowrap; width: 15px;" title="Enlarge"></a></div>Derivation of pentatope numbers from a left-justified <a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle" style="background: none; color: #3366cc; text-decoration-line: none;" title="Pascal's triangle">Pascal's triangle</a></div></div></div><p style="margin: 0.5em 0px;">A <b><a class="mw-redirect" href="https://en.wikipedia.org/wiki/Pentatope" style="background: none; color: #3366cc; text-decoration-line: none;" title="Pentatope">pentatope</a> number</b> is a number in the fifth cell of any row of <a href="https://en.wikipedia.org/wiki/Pascal%27s_triangle" style="background: none; color: #3366cc; text-decoration-line: none;" title="Pascal's triangle">Pascal's triangle</a> starting with the 5-term row <span class="nowrap" style="white-space: nowrap;">1 4 6 4 1</span>, either from left to right or from right to left.</p><p style="margin: 0.5em 0px;">The first few numbers of this kind are:</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><a class="mw-redirect" href="https://en.wikipedia.org/wiki/1_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="1 (number)">1</a>, <a class="mw-redirect" href="https://en.wikipedia.org/wiki/5_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="5 (number)">5</a>, <a href="https://en.wikipedia.org/wiki/15_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="15 (number)">15</a>, <a href="https://en.wikipedia.org/wiki/35_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="35 (number)">35</a>, <a href="https://en.wikipedia.org/wiki/70_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="70 (number)">70</a>, <a href="https://en.wikipedia.org/wiki/126_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="126 (number)">126</a>, 210, 330, <a href="https://en.wikipedia.org/wiki/495_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="495 (number)">495</a>, 715, <a href="https://en.wikipedia.org/wiki/1001_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="1001 (number)">1001</a>, 1365 (sequence <span class="nowrap external" style="white-space: nowrap;"><a class="extiw" href="https://oeis.org/A000332" style="background: none; color: #3366cc; text-decoration-line: none;" title="oeis:A000332">A000332</a></span> in the <a href="https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences" style="background: none; color: #3366cc; text-decoration-line: none;" title="On-Line Encyclopedia of Integer Sequences">OEIS</a>)</dd></dl><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin: 0.5em 0px 1.3em 1.4em; width: auto;"><div class="thumbinner" style="background-color: #f8f9fa; border: 1px solid rgb(200, 204, 209); font-size: 13.16px; min-width: 100px; overflow: hidden; padding: 3px; text-align: center; width: 162px;"><a class="image" href="https://en.wikipedia.org/wiki/File:Pentatope_of_70_spheres_animation.gif" style="background: none; color: #3366cc; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="120" data-file-width="160" decoding="async" height="120" src="https://upload.wikimedia.org/wikipedia/commons/0/0f/Pentatope_of_70_spheres_animation.gif" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="160" /></a><div class="thumbcaption" style="border: 0px; font-size: 12.3704px; line-height: 1.4em; padding: 3px; text-align: left;">A <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Pentatope" style="background: none; color: #3366cc; text-decoration-line: none;" title="Pentatope">pentatope</a> with side length 5 contains 70 <a href="https://en.wikipedia.org/wiki/3-sphere" style="background: none; color: #3366cc; text-decoration-line: none;" title="3-sphere">3-spheres</a>. Each layer represents one of the first five <a href="https://en.wikipedia.org/wiki/Tetrahedral_number" style="background: none; color: #3366cc; text-decoration-line: none;" title="Tetrahedral number">tetrahedral numbers</a>. For example, the bottom (green) layer has 35 <a href="https://en.wikipedia.org/wiki/Sphere" style="background: none; color: #3366cc; text-decoration-line: none;" title="Sphere">spheres</a> in total.</div></div></div><p style="margin: 0.5em 0px;">Pentatope numbers belong to the class of <a href="https://en.wikipedia.org/wiki/Figurate_number" style="background: none; color: #3366cc; text-decoration-line: none;" title="Figurate number">figurate numbers</a>, which can be represented as regular, discrete geometric patterns.<sup class="reference" id="cite_ref-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Pentatope_number#cite_note-1" style="background: none; color: #3366cc; text-decoration-line: none;">[1]</a></sup></p><h2 style="border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span class="mw-headline" id="Formula">Formula</span><span class="mw-editsection" face="sans-serif" style="font-size: small; line-height: 0; margin-left: 1em; margin-right: 0px; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Pentatope_number&action=edit&section=1" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Edit section: Formula">edit</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h2><p style="margin: 0.5em 0px;">The formula for the <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span>th pentatope number is represented by the 4th <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Rising_factorial" style="background: none; color: #3366cc; text-decoration-line: none;" title="Rising factorial">rising factorial</a> of <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span> divided by the <a href="https://en.wikipedia.org/wiki/Factorial" style="background: none; color: #3366cc; text-decoration-line: none;" title="Factorial">factorial</a> of 4:</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle P_{n}={\frac {n!}{4!(n-4)!}}={\frac {n(n+1)(n+2)(n+3)}{24}}.}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>�</mi><mo>!</mo></mrow><mrow><mn>4</mn><mo>!</mo><mo stretchy="false">(</mo><mi>�</mi><mo>−</mo><mn>4</mn><mo stretchy="false">)</mo><mo>!</mo></mrow></mfrac></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><mn>24</mn></mfrac></mrow><mo>.</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle P_{n}={\frac {n!}{4!(n-4)!}}={\frac {n(n+1)(n+2)(n+3)}{24}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7f93b8be3c20bfaff683bed878dd4cfbad8cdb87" style="border: 0px; display: inline-block; height: 6.509ex; vertical-align: -2.671ex; width: 43.905ex;" /></span></dd></dl><p style="margin: 0.5em 0px;">The pentatope numbers can also be represented as <a href="https://en.wikipedia.org/wiki/Binomial_coefficient" style="background: none; color: #3366cc; text-decoration-line: none;" title="Binomial coefficient">binomial coefficients</a>:</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle P_{n}={\binom {n+3}{4}},}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mrow><mi>�</mi><mo>+</mo><mn>3</mn></mrow><mn>4</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow></mrow><mo>,</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle P_{n}={\binom {n+3}{4}},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f136b50dcf71353c8b1d50e57f327a812bfc1d52" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 15.275ex;" /></span></dd></dl><p style="margin: 0.5em 0px;">which is the number of distinct <a class="mw-redirect" href="https://en.wikipedia.org/wiki/4-tuple" style="background: none; color: #3366cc; text-decoration-line: none;" title="4-tuple">quadruples</a> that can be selected from <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>n</i> + 3</span> objects, and it is read aloud as "<span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;"><i>n</i></span> plus three choose four".</p><h2 style="border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span class="mw-headline" id="Properties">Properties</span><span class="mw-editsection" face="sans-serif" style="font-size: small; line-height: 0; margin-left: 1em; margin-right: 0px; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Pentatope_number&action=edit&section=2" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Edit section: Properties">edit</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h2><p style="margin: 0.5em 0px;">Two of every three pentatope numbers are also <a href="https://en.wikipedia.org/wiki/Pentagonal_number" style="background: none; color: #3366cc; text-decoration-line: none;" title="Pentagonal number">pentagonal numbers</a>. To be precise, the <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">(3<i>k</i> − 2)</span>th pentatope number is always the <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">(<span class="sfrac tion" role="math" style="display: inline-block; font-size: 14.042px; text-align: center; vertical-align: -0.5em;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">3<i>k</i><sup style="font-size: 11.2336px; line-height: 1;">2</sup> − <i>k</i></span><span class="sr-only" style="border: 0px; clip: rect(0px, 0px, 0px, 0px); height: 1px; margin: -1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span>)</span>th pentagonal number and the <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">(3<i>k</i> − 1)</span>th pentatope number is always the <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">(<span class="sfrac tion" role="math" style="display: inline-block; font-size: 14.042px; text-align: center; vertical-align: -0.5em;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">3<i>k</i><sup style="font-size: 11.2336px; line-height: 1;">2</sup> + <i>k</i></span><span class="sr-only" style="border: 0px; clip: rect(0px, 0px, 0px, 0px); height: 1px; margin: -1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span>)</span>th pentagonal number. The <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">(3<i>k</i>)</span>th pentatope number is the <a href="https://en.wikipedia.org/wiki/Pentagonal_number" style="background: none; color: #3366cc; text-decoration-line: none;" title="Pentagonal number">generalized pentagonal number</a> obtained by taking the negative index <span class="texhtml" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">−<span class="sfrac tion" role="math" style="display: inline-block; font-size: 14.042px; text-align: center; vertical-align: -0.5em;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">3<i>k</i><sup style="font-size: 11.2336px; line-height: 1;">2</sup> + <i>k</i></span><span class="sr-only" style="border: 0px; clip: rect(0px, 0px, 0px, 0px); height: 1px; margin: -1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">2</span></span></span> in the formula for pentagonal numbers. (These expressions always give <a href="https://en.wikipedia.org/wiki/Integer" style="background: none; color: #3366cc; text-decoration-line: none;" title="Integer">integers</a>).<sup class="reference" id="cite_ref-oeis_2-0" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Pentatope_number#cite_note-oeis-2" style="background: none; color: #3366cc; text-decoration-line: none;">[2]</a></sup></p><p style="margin: 0.5em 0px;"><b><span style="font-size: large;">The <a class="mw-redirect" href="https://en.wikipedia.org/wiki/Infinite_sum" style="background: none; color: #3366cc; text-decoration-line: none;" title="Infinite sum">infinite sum</a> of the <a href="https://en.wikipedia.org/wiki/Multiplicative_inverse" style="background: none; color: #3366cc; text-decoration-line: none;" title="Multiplicative inverse">reciprocals</a> of all pentatope numbers is <span class="sfrac tion" role="math" style="display: inline-block; text-align: center; vertical-align: -0.5em; white-space: nowrap;"><span class="num" style="display: block; line-height: 1em; margin: 0px 0.1em;">4</span><span class="sr-only" style="border: 0px; clip: rect(0px, 0px, 0px, 0px); height: 1px; margin: -1px; overflow: hidden; padding: 0px; position: absolute; width: 1px;">/</span><span class="den" style="border-top: 1px solid; display: block; line-height: 1em; margin: 0px 0.1em;">3</span></span>.<sup class="reference" id="cite_ref-3" style="line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Pentatope_number#cite_note-3" style="background: none; color: #3366cc; text-decoration-line: none;">[3]</a></sup> This can be derived using <a href="https://en.wikipedia.org/wiki/Telescoping_series" style="background: none; color: #3366cc; text-decoration-line: none;" title="Telescoping series">telescoping series</a>.</span></b></p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle \sum _{n=1}^{\infty }{\frac {4!}{n(n+1)(n+2)(n+3)}}={\frac {4}{3}}.}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="normal">∞</mi></mrow></munderover><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mn>4</mn><mo>!</mo></mrow><mrow><mi>�</mi><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo></mrow></mfrac></mrow><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mn>4</mn><mn>3</mn></mfrac></mrow><mo>.</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle \sum _{n=1}^{\infty }{\frac {4!}{n(n+1)(n+2)(n+3)}}={\frac {4}{3}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/31263448269517483db09cf6b767ad76ca1cdb8d" style="border: 0px; display: inline-block; height: 6.843ex; vertical-align: -3.005ex; width: 33.337ex;" /></span></dd></dl><p style="margin: 0.5em 0px;">Pentatope numbers can be represented as the sum of the first <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span> <a href="https://en.wikipedia.org/wiki/Tetrahedral_number" style="background: none; color: #3366cc; text-decoration-line: none;" title="Tetrahedral number">tetrahedral numbers</a>:<sup class="reference" id="cite_ref-oeis_2-1" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Pentatope_number#cite_note-oeis-2" style="background: none; color: #3366cc; text-decoration-line: none;">[2]</a></sup></p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle P_{n}=\sum _{k=1}^{n}Te_{n},}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msub><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></munderover><mi>�</mi><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msub><mo>,</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle P_{n}=\sum _{k=1}^{n}Te_{n},}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8a236759b5df4183a543e845f3dd669e2724d2a4" style="border: 0px; display: inline-block; height: 6.843ex; vertical-align: -3.005ex; width: 14.136ex;" /></span></dd></dl><p style="margin: 0.5em 0px;">and are also related to tetrahedral numbers themselves:</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle P_{n}={\tfrac {1}{4}}(n+3)Te_{n}.}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msub><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="false" scriptlevel="0"><mfrac><mn>1</mn><mn>4</mn></mfrac></mstyle></mrow><mo stretchy="false">(</mo><mi>�</mi><mo>+</mo><mn>3</mn><mo stretchy="false">)</mo><mi>�</mi><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></msub><mo>.</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle P_{n}={\tfrac {1}{4}}(n+3)Te_{n}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606313515c64a322829a820025c6a0062fe7659a" style="border: 0px; display: inline-block; height: 3.509ex; vertical-align: -1.171ex; width: 19.259ex;" /></span></dd></dl><p style="margin: 0.5em 0px;">No <a href="https://en.wikipedia.org/wiki/Prime_number" style="background: none; color: #3366cc; text-decoration-line: none;" title="Prime number">prime number</a> is the predecessor of a pentatope number (it needs to check only -1 and 4=2<sup style="font-size: 11.2px; line-height: 1;">2</sup>), and the largest <a href="https://en.wikipedia.org/wiki/Semiprime" style="background: none; color: #3366cc; text-decoration-line: none;" title="Semiprime">semiprime</a> which is the predecessor of a pentatope number is 1819.</p><p style="margin: 0.5em 0px;">Similarly, the only primes preceding a <a href="https://en.wikipedia.org/wiki/Figurate_number#Triangular_numbers_and_their_analogs_in_higher_dimensions" style="background: none; color: #3366cc; text-decoration-line: none;" title="Figurate number">6-simplex number</a> are <a href="https://en.wikipedia.org/wiki/83_(number)" style="background: none; color: #3366cc; text-decoration-line: none;" title="83 (number)">83</a> and 461.</p><h2 style="border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span class="mw-headline" id="Test_for_pentatope_numbers">Test for pentatope numbers</span><span class="mw-editsection" face="sans-serif" style="font-size: small; line-height: 0; margin-left: 1em; margin-right: 0px; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Pentatope_number&action=edit&section=3" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Edit section: Test for pentatope numbers">edit</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h2><p style="margin: 0.5em 0px;">We can derive this test from the formula for the <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span>th pentatope number.</p><p style="margin: 0.5em 0px;">Given a positive integer <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">x</span>, to test whether it is a pentatope number we can compute</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle n={\frac {{\sqrt {5+4{\sqrt {24x+1}}}}-3}{2}}.}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mi>�</mi><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mfrac><mrow><mrow class="MJX-TeXAtom-ORD"><msqrt><mn>5</mn><mo>+</mo><mn>4</mn><mrow class="MJX-TeXAtom-ORD"><msqrt><mn>24</mn><mi>�</mi><mo>+</mo><mn>1</mn></msqrt></mrow></msqrt></mrow><mo>−</mo><mn>3</mn></mrow><mn>2</mn></mfrac></mrow><mo>.</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle n={\frac {{\sqrt {5+4{\sqrt {24x+1}}}}-3}{2}}.}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f29e120a2bd8756a12d536aa7a1be66c9ac6ffa7" style="border: 0px; display: inline-block; height: 7.676ex; vertical-align: -1.838ex; width: 27.061ex;" /></span><sup class="noprint Inline-Template Template-Fact" style="font-size: 11.2px; line-height: 1; white-space: nowrap;">[<i><a href="https://en.wikipedia.org/wiki/Wikipedia:Citation_needed" style="background: none; color: #3366cc; text-decoration-line: none;" title="Wikipedia:Citation needed"><span title="A source is needed for this test to verify that is is actually effective; one would think that the rounding errors introduced when using a computer to calculate the nested square roots would cause the result to be very close to an integer (but not exactly an integer) when x is a pentatope number. (March 2021)">citation needed</span></a></i>]</sup></dd></dl><p style="margin: 0.5em 0px;">The number <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">x</span> is pentatope if and only if <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span> is a <a href="https://en.wikipedia.org/wiki/Natural_number" style="background: none; color: #3366cc; text-decoration-line: none;" title="Natural number">natural number</a>. In that case <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">x</span> is the <span class="texhtml mvar" style="font-family: "Nimbus Roman No9 L", "Times New Roman", Times, serif; font-feature-settings: "lnum", "tnum", "kern" 0; font-kerning: none; font-size: 16.52px; font-style: italic; font-variant-numeric: lining-nums tabular-nums; line-height: 1; white-space: nowrap;">n</span>th pentatope number.</p><h2 style="border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span class="mw-headline" id="Generating_function">Generating function</span><span class="mw-editsection" face="sans-serif" style="font-size: small; line-height: 0; margin-left: 1em; margin-right: 0px; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Pentatope_number&action=edit&section=4" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Edit section: Generating function">edit</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h2><p style="margin: 0.5em 0px;">The <a href="https://en.wikipedia.org/wiki/Generating_function" style="background: none; color: #3366cc; text-decoration-line: none;" title="Generating function">generating function</a> for pentatope numbers is<sup class="reference" id="cite_ref-4" style="font-size: 11.2px; line-height: 1; unicode-bidi: isolate; white-space: nowrap;"><a href="https://en.wikipedia.org/wiki/Pentatope_number#cite_note-4" style="background: none; color: #3366cc; text-decoration-line: none;">[4]</a></sup></p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; font-size: 16.52px; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle {\frac {x}{(1-x)^{5}}}=x+5x^{2}+15x^{3}+35x^{4}+\dots .}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mrow class="MJX-TeXAtom-ORD"><mfrac><mi>�</mi><mrow><mo stretchy="false">(</mo><mn>1</mn><mo>−</mo><mi>�</mi><msup><mo stretchy="false">)</mo><mrow class="MJX-TeXAtom-ORD"><mn>5</mn></mrow></msup></mrow></mfrac></mrow><mo>=</mo><mi>�</mi><mo>+</mo><mn>5</mn><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msup><mo>+</mo><mn>15</mn><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msup><mo>+</mo><mn>35</mn><msup><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msup><mo>+</mo><mo>…</mo><mo>.</mo></mstyle></mrow></semantics></math></span><img alt="{\displaystyle {\frac {x}{(1-x)^{5}}}=x+5x^{2}+15x^{3}+35x^{4}+\dots .}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/faaa6e5a64530f54b1eed9820d0638e091352e22" style="border: 0px; display: inline-block; height: 5.509ex; vertical-align: -2.671ex; width: 41.543ex;" /></span></dd></dl><h2 style="border-bottom: 1px solid rgb(162, 169, 177); color: black; font-family: "Linux Libertine", Georgia, Times, serif; font-weight: normal; line-height: 1.3; margin: 1em 0px 0.25em; overflow: hidden; padding: 0px;"><span class="mw-headline" id="Applications">Applications</span><span class="mw-editsection" face="sans-serif" style="font-size: small; line-height: 0; margin-left: 1em; margin-right: 0px; unicode-bidi: isolate; user-select: none; vertical-align: baseline;"><span class="mw-editsection-bracket" style="color: #54595d; margin-right: 0.25em;">[</span><a href="https://en.wikipedia.org/w/index.php?title=Pentatope_number&action=edit&section=5" style="background: none; color: #3366cc; text-decoration-line: none; white-space: nowrap;" title="Edit section: Applications">edit</a><span class="mw-editsection-bracket" style="color: #54595d; margin-left: 0.25em;">]</span></span></h2><p style="margin: 0.5em 0px;">In <a href="https://en.wikipedia.org/wiki/Biochemistry" style="background: none; color: #3366cc; text-decoration-line: none;" title="Biochemistry">biochemistry</a>, the pentatope numbers represent the number of possible arrangements of <i>n</i> different polypeptide subunits in a tetrameric (tetrahedral) protein.</p><p style="margin: 0.5em 0px;"><br /></p><header class="mw-body-header vector-page-titlebar" style="align-items: center; box-shadow: rgb(162, 169, 177) 0px 1px; display: flex; font-size: 16px; justify-content: flex-end;"><h1 class="firstHeading mw-first-heading" id="firstHeading" style="border: 0px; color: black; flex-grow: 1; font-family: "Linux Libertine", Georgia, Times, serif; font-size: 1.8em; font-weight: normal; line-height: 1.3; margin: 0px; overflow-wrap: break-word; overflow: hidden; padding: 0px;"><span class="mw-page-title-main">Nombre pentatopique</span></h1><div class="vector-menu vector-dropdown vector-menu-dropdown mw-portlet mw-portlet-lang mw-ui-icon-flush-right" id="p-lang-btn" style="box-sizing: border-box; flex-shrink: 0; float: right; 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color: #3366cc; text-decoration-line: none;" title="Projet:Accueil">projets correspondants</a>.</p></div></div><div class="thumb tright" style="background-color: transparent; clear: right; float: right; margin: 0.5em 0px 1.3em 1.4em; width: auto;"><div class="thumbinner" style="background-color: #f8f9fa; border: 1px solid rgb(200, 204, 209); font-size: 13.16px; min-width: 100px; overflow: hidden; padding: 3px; text-align: center; width: 162px;"><a class="image" href="https://commons.wikimedia.org/wiki/File:Pentatope_of_70_spheres_animation.gif?uselang=fr" style="background: none; color: #3366cc; text-decoration-line: none;"><img alt="" class="thumbimage" data-file-height="120" data-file-width="160" decoding="async" height="120" src="https://upload.wikimedia.org/wikipedia/commons/0/0f/Pentatope_of_70_spheres_animation.gif" style="background-color: white; border: 1px solid rgb(200, 204, 209); vertical-align: middle;" width="160" /></a><div class="thumbcaption" style="border: 0px; font-size: 12.3704px; line-height: 1.4em; padding: 3px; text-align: left;">Un pentatope à 70 sphères. Chaque niveau représente un des 5 premiers <a href="https://fr.wikipedia.org/wiki/Nombre_t%C3%A9tra%C3%A9drique" style="background: none; color: #3366cc; text-decoration-line: none;" title="Nombre tétraédrique">nombres tétraédriques</a>. Par exemple, le niveau vert possède 35 sphères en tout.</div></div></div><p style="margin: 0.5em 0px;">Un <b>nombre pentatopique</b> est un nombre de la cinquième diagonale descendante du <a href="https://fr.wikipedia.org/wiki/Triangle_de_Pascal" style="background: none; color: #3366cc; text-decoration-line: none;" title="Triangle de Pascal">triangle de Pascal</a>. Les premiers nombres de cette sorte sont 1, 5, 15, 35, 70, et 126.</p><p style="margin: 0.5em 0px;">Les nombres pentatopiques sont des <a href="https://fr.wikipedia.org/wiki/Nombre_figur%C3%A9" style="background: none; color: #3366cc; text-decoration-line: none;" title="Nombre figuré">nombres figurés</a>. Ils peuvent idéalement être représentés en dimension 4 par un <a href="https://fr.wikipedia.org/wiki/Polytope" style="background: none; color: #3366cc; text-decoration-line: none;" title="Polytope">polytope</a> constitué d'un empilement de <a href="https://fr.wikipedia.org/wiki/T%C3%A9tra%C3%A8dre" style="background: none; color: #3366cc; text-decoration-line: none;" title="Tétraèdre">tétraèdres</a> réguliers.</p><p style="margin: 0.5em 0px;">Le nombre pentatopique de rang <i>n</i> est donc la somme des <i>n</i> premiers nombres tétraédriques</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle P_{4}(n)=\sum _{k=1}^{n}P_{3}(n)=\sum _{k=1}^{n}{n+2 \choose 3}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msub><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></munderover><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msub><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>=</mo><munderover><mo>∑</mo><mrow class="MJX-TeXAtom-ORD"><mi>�</mi><mo>=</mo><mn>1</mn></mrow><mrow class="MJX-TeXAtom-ORD"><mi>�</mi></mrow></munderover><mrow class="MJX-TeXAtom-ORD"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mrow><mi>�</mi><mo>+</mo><mn>2</mn></mrow><mn>3</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow></mrow></mstyle></mrow></semantics></math></span><img alt="{\displaystyle P_{4}(n)=\sum _{k=1}^{n}P_{3}(n)=\sum _{k=1}^{n}{n+2 \choose 3}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/36356a281b824999b1fe9238fba3c660565c3527" style="border: 0px; display: inline-block; height: 6.843ex; vertical-align: -3.005ex; width: 34.001ex;" /></span></dd></dl><p style="margin: 0.5em 0px;">On obtient donc la formule</p><dl style="margin-bottom: 0.5em; margin-top: 0.2em;"><dd style="margin-bottom: 0.1em; margin-left: 1.6em; margin-right: 0px;"><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="clip: rect(1px, 1px, 1px, 1px); display: none; height: 1px; opacity: 0; overflow: hidden; position: absolute; width: 1px;"><math alttext="{\displaystyle P_{4}(n)={n+3 \choose 4}}" xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><msub><mi>�</mi><mrow class="MJX-TeXAtom-ORD"><mn>4</mn></mrow></msub><mo stretchy="false">(</mo><mi>�</mi><mo stretchy="false">)</mo><mo>=</mo><mrow class="MJX-TeXAtom-ORD"><mrow><mrow class="MJX-TeXAtom-OPEN"><mo maxsize="2.047em" minsize="2.047em">(</mo></mrow><mfrac linethickness="0"><mrow><mi>�</mi><mo>+</mo><mn>3</mn></mrow><mn>4</mn></mfrac><mrow class="MJX-TeXAtom-CLOSE"><mo maxsize="2.047em" minsize="2.047em">)</mo></mrow></mrow></mrow></mstyle></mrow></semantics></math></span><img alt="{\displaystyle P_{4}(n)={n+3 \choose 4}}" aria-hidden="true" class="mwe-math-fallback-image-inline" src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b01559b7e0b05ddd3dcf55d225d9b0e6a5839fc7" style="border: 0px; display: inline-block; height: 6.176ex; vertical-align: -2.505ex; width: 17.668ex;" /></span></dd></dl><p style="margin: 0.5em 0px;">Il n'est donc pas surprenant de les rencontrer dans la cinquième diagonale du triangle de Pascal.</p></div></div></div></div></div></div>alain planchonhttp://www.blogger.com/profile/17041209264618533294noreply@blogger.com0