List of repunit primes

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 $$n$$ 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 ($$11_4$$). $$4^n-1 = \left(2^n+1\right)\left(2^n-1\right)$$, and 3 always divides $$2^n + 1$$ when n is odd and $$2^n - 1$$ when n is even. For n greater than 2, both $$2^n + 1$$ and $$2^n - 1$$ 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 $$n$$ of
 * 3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593, 4939471, ....

Base 6 repunit primes
The first few base-6 repunit primes are
 * 7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, 133733063818254349335501779590081460423013416258060407531857720755181857441961908284738707408499507 ,

corresponding to $$n$$ 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 $$n$$ of
 * 5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ....

Base 8 repunit primes
The only base-8 repunit prime is 73 ($$111_8$$). $$8^n-1 = \left(4^n+2^n+1\right)\left(2^n-1\right)$$, and 7 always divides $$4^n + 2^n + 1$$ when n is not divisible by 3 and $$2^n - 1$$ when n is divisible by 3. For n greater than 3, both $$4^n + 2^n + 1$$ and $$2^n - 1$$ 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. $$9^n-1 = \left(3^n+1\right)\left(3^n-1\right)$$, and $$3^n + 1$$ and $$3^n - 1$$ are even, and one of $$3^n + 1$$ and $$3^n - 1$$ is divisible by 4. For n greater than 1, both $$3^n + 1$$ and $$3^n - 1$$ are greater than 4, so removing the factor of 8 (which is equivalent to removing the factor 4 from $$3^n + 1$$ or $$3^n - 1$$, and removing the factor 2 from the other number) still leaves two factors greater than 1. Therefore, the number cannot be prime.

Base 11 repunit primes
The first few base-11 repunit primes are
 * 50544702849929377, 6115909044841454629, 1051153199500053598403188407217590190707671147285551702341089650185945215953, 567000232521795739625828281267171344486805385881217575081149660163046217465544573355710592079769932651989153833612198334843467861091902034340949

corresponding to $$n$$ 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 $$n$$ 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 ($$11_{16}$$). $$16^n-1 = \left(4^n+1\right)\left(4^n-1\right)$$, and 3 always divides $$4^n - 1$$, and 5 always divides $$4^n + 1$$ when n is odd and $$4^n - 1$$ when n is even. For n greater than 2, both $$4^n + 1$$ and $$4^n - 1$$ 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 $$n$$ of
 * 3, 11, 17, 1487, 31013, 48859, 61403, 472709, 984349, ....

Bases $$b$$ such that $$R_p(b)$$ is prime for prime $$p$$
Smallest base $$b$$ such that $$R_p(b)$$ is prime (where $$p$$ is the $$n$$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 $$b$$ such that $$R_p(-b)$$ is prime (where $$p$$ is the $$n$$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, ...

List of repunit primes base $$b$$
Smallest prime $$p>2$$ such that $$R_p(b)$$ is prime are (start with $$b=2$$, 0 if no such $$p$$ exists)


 * 3, 3, 0, 3, 3, 5, 3, 0, 19, 17, 3, 5, 3, 3, 0, 3, 25667, 19, 3, 3, 5, 5, 3, 0, 7, 3, 5, 5, 5, 7, 0, 3, 13, 313, 0, 13, 3, 349, 5, 3, 1319, 5, 5, 19, 7, 127, 19, 0, 3, 4229, 103, 11, 3, 17, 7, 3, 41, 3, 7, 7, 3, 5, 0, 19, 3, 19, 5, 3, 29, 3, 7, 5, 5, 3, 41, 3, 3, 5, 3, 0, 23, 5, 17, 5, 11, 7, 61, 3, 3, 4421, 439, 7, 5, 7, 3343, 17, 13, 3, 0, 3, ...

Smallest prime $$p>2$$ such that $$R_p(-b)$$ is prime are (start with $$b=2$$, 0 if no such $$p$$ exists)


 * 3, 3, 3, 5, 3, 3, 0, 3, 5, 5, 5, 3, 7, 3, 3, 7, 3, 17, 5, 3, 3, 11, 7, 3, 11, 0, 3, 7, 139, 109, 0, 5, 3, 11, 31, 5, 5, 3, 53, 17, 3, 5, 7, 103, 7, 5, 5, 7, 1153, 3, 7, 21943, 7, 3, 37, 53, 3, 17, 3, 7, 11, 3, 0, 19, 7, 3, 757, 11, 3, 5, 3, 7, 13, 5, 3, 37, 3, 3, 5, 3, 293, 19, 7, 167, 7, 7, 709, 13, 3, 3, 37, 89, 71, 43, 37, (>500000), 19, 7, 3, 7, ...


 * Repunits with negative base and even n are negative. If their absolute value is prime then they are included above and marked with an asterisk. They are not included in the corresponding OEIS sequences.

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