This is a list of repunit primes in various bases.
Base 2 repunit primes
Base-2 repunit primes are called Mersenne primes.
Base 3 repunit primes The first few base-3 repunit primes are 13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 , corresponding to of 3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117, ... .
Base 4 repunit primes The only base-4 repunit prime is 5 (). , and 3 always divides when n is odd and when n is even. For n greater than 2, both and are greater than 3, so removing the factor of 3 still leaves two factors greater than 1. Therefore, the number cannot be prime.
Base 5 repunit primes The first few base-5 repunit primes are 31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, 35032461608120426773093239582247903282006548546912894293926707097244777067146515037165954709053039550781, 815663058499815565838786763657068444462645532258620818469829556933715405574685778402862015856733535201783524826169013977050781 , corresponding to of 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593, ..., 4939471, ..., 5154509, ... .
Base 6 repunit primes The first few base-6 repunit primes are 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, 133733063818254349335501779590081460423013416258060407531857720755181857441961908284738707408499507 , corresponding to of 2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, 1365019, 3360347, ... .
Base 7 repunit primes The first few base-7 repunit primes are 2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601 corresponding to of 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... .
Base 8 repunit primes The only base-8 repunit prime is 73 (). , and 7 always divides when n is not divisible by 3 and when n is divisible by 3. For n greater than 3, both and are greater than 7, so removing the factor of 7 still leaves two factors greater than 1. Therefore, the number cannot be prime.
Base 9 repunit primes There are no base-9 repunit primes. , and and are even, and one of and is divisible by 4. For n greater than 1, both and are greater than 4, so removing the factor of 8 (which is equivalent to removing the factor 4 from or , and removing the factor 2 from the other number) still leaves two factors greater than 1. Therefore, the number cannot be prime.
Base 10 repunit primes
Base 11 repunit primes The first few base-11 repunit primes are 50544702849929377, 6115909044841454629, 1051153199500053598403188407217590190707671147285551702341089650185945215953, 567000232521795739625828281267171344486805385881217575081149660163046217465544573355710592079769932651989153833612198334843467861091902034340949 corresponding to of 17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, 1868983, ... .
Base 12 repunit primes The first few base-12 repunit primes are 13, 157, 22621, 29043636306420266077, 43570062353753446053455610056679740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941 corresponding to of 2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... .
Base 16 repunit primes The only base-16 repunit prime is 17 (). , and 3 always divides , and 5 always divides when n is odd and when n is even. For n greater than 2, both and are greater than 15, so removing the factor of 15 still leaves two factors greater than 1. Therefore, the number cannot be prime.
Base 20 repunit primes The first few base-20 repunit primes are 421, 10778947368421, 689852631578947368421 corresponding to of 3, 11, 17, 1487, 31013, 48859, 61403, 472709, 984349, ... .
Bases b such that Rp(b) is prime for prime p
Smallest base such that is prime (where is the th prime) are
2, 2, 2, 2, 5, 2, 2, 2, 10, 6, 2, 61, 14, 15, 5, 24, 19, 2, 46, 3, 11, 22, 41, 2, 12, 22, 3, 2, 12, 86, 2, 7, 13, 11, 5, 29, 56, 30, 44, 60, 304, 5, 74, 118, 33, 156, 46, 183, 72, 606, 602, 223, 115, 37, 52, 104, 41, 6, 338, 217, 13, 136, 220, 162, 35, 10, 218, 19, 26, 39, 12, 22, 67, 120, 195, 48, 54, 463, 38, 41, 17, 808, 404, 46, 76, 793, 38, 28, 215, 37, 236, 59, 15, 514, 260, 498, 6, 2, 95, 3, ...
Smallest base such that is prime (where is the th prime) are
3, 2, 2, 2, 2, 2, 2, 2, 2, 7, 2, 16, 61, 2, 6, 10, 6, 2, 5, 46, 18, 2, 49, 16, 70, 2, 5, 6, 12, 92, 2, 48, 89, 30, 16, 147, 19, 19, 2, 16, 11, 289, 2, 12, 52, 2, 66, 9, 22, 5, 489, 69, 137, 16, 36, 96, 76, 117, 26, 3, 159, 10, 16, 209, 2, 16, 23, 273, 2, 460, 22, 3, 36, 28, 329, 43, 69, 86, 271, 396, 28, 83, 302, 209, 11, 300, 159, 79, 31, 331, 52, 176, 3, 28, 217, 14, 410, 252, 718, 164, ...