The term repunit comes from the words 'repeated' and 'unit;' so repunits are positive integers in which every digit is one. (This term was coined by A. H. Beiler in [Beiler1964].)
For example, R1=1, R2=11, R3=111, and Rn=(10n-1)/9.
Notice Rn divides Rm whenever n divides m.
Repunit primes are repunits that are prime.
For example, 11, 1111111111111111111, and 11111111111111111111111 (2, 19, and 23 digits). The only other known
repunit primes are the ones with 317 digits: (10
317-1)/9, 1,031 digits and (10
1031-1)/9.
Factorization of decimal repunits
(Prime factors colored red means "new factors", i. e. the prime factor divides Rn but does not divide Rk for all k < n) (sequence A102380 in the OEIS)[2]
| R1 = | 1 | | R2 = | 11 | | R3 = | 3 · 37 | | R4 = | 11 · 101 | | R5 = | 41 · 271 | | R6 = | 3 · 7 · 11 · 13 · 37 | | R7 = | 239 · 4649 | | R8 = | 11 · 73 · 101 · 137 | | R9 = | 32 · 37 · 333667 | | R10 = | 11 · 41 · 271 · 9091 |
| | R11 = | 21649 · 513239 | | R12 = | 3 · 7 · 11 · 13 · 37 · 101 · 9901 | | R13 = | 53 · 79 · 265371653 | | R14 = | 11 · 239 · 4649 · 909091 | | R15 = | 3 · 31 · 37 · 41 · 271 · 2906161 | | R16 = | 11 · 17 · 73 · 101 · 137 · 5882353 | | R17 = | 2071723 · 5363222357 | | R18 = | 32 · 7 · 11 · 13 · 19 · 37 · 52579 · 333667 | | R19 = | 1111111111111111111 | | R20 = | 11 · 41 · 101 · 271 · 3541 · 9091 · 27961 |
| | R21 = | 3 · 37 · 43 · 239 · 1933 · 4649 · 10838689 | | R22 = | 112 · 23 · 4093 · 8779 · 21649 · 513239 | | R23 = | 11111111111111111111111 | | R24 = | 3 · 7 · 11 · 13 · 37 · 73 · 101 · 137 · 9901 · 99990001 | | R25 = | 41 · 271 · 21401 · 25601 · 182521213001 | | R26 = | 11 · 53 · 79 · 859 · 265371653 · 1058313049 | | R27 = | 33 · 37 · 757 · 333667 · 440334654777631 | | R28 = | 11 · 29 · 101 · 239 · 281 · 4649 · 909091 · 121499449 | | R29 = | 3191 · 16763 · 43037 · 62003 · 77843839397 | | R30 = | 3 · 7 · 11 · 13 · 31 · 37 · 41 · 211 · 241 · 271 · 2161 · 9091 · 2906161 |
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Smallest prime factor of Rn for n > 1 are
- 11, 3, 11, 41, 3, 239, 11, 3, 11, 21649, 3, 53, 11, 3, 11, 2071723, 3, 1111111111111111111, 11, 3, 11, 11111111111111111111111, 3, 41, 11, 3, 11, 3191, 3, 2791, 11, 3, 11, 41, 3, 2028119, 11, 3, 11, 83, 3, 173, 11, 3, 11, 35121409, 3, 239, 11, ... (sequence A067063 in the OEIS)
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