List of repunit primes

This is a list of repunit primes in various bases.

Base-2 repunit primes are called Mersenne primes.

The first few base-3 repunit primes are

13, 1093, 797161, 3754733257489862401973357979128773, 6957596529882152968992225251835887181478451547013 (sequence A076481 in the OEIS),

corresponding to ${\displaystyle n}$of

3, 7, 13, 71, 103, 541, 1091, 1367, 1627, 4177, 9011, 9551, 36913, 43063, 49681, 57917, 483611, 877843, 2215303, 2704981, 3598867, 7973131, 8530117, ... (sequence A028491 in the OEIS).

The only base-4 repunit prime is 5 (${\displaystyle 11_{4}}$). ${\displaystyle 4^{n}-1=\left(2^{n}+1\right)\left(2^{n}-1\right)}$, and 3 always divides ${\displaystyle 2^{n}+1}$when n is odd and ${\displaystyle 2^{n}-1}$when n is even. For n greater than 2, both ${\displaystyle 2^{n}+1}$and ${\displaystyle 2^{n}-1}$are greater than 3, so removing the factor of 3 still leaves two factors greater than 1. Therefore, the number cannot be prime.

The first few base-5 repunit primes are

31, 19531, 12207031, 305175781, 177635683940025046467781066894531, 14693679385278593849609206715278070972733319459651094018859396328480215743184089660644531, 35032461608120426773093239582247903282006548546912894293926707097244777067146515037165954709053039550781, 815663058499815565838786763657068444462645532258620818469829556933715405574685778402862015856733535201783524826169013977050781 (sequence A086122 in the OEIS),

corresponding to ${\displaystyle n}$of

3, 7, 11, 13, 47, 127, 149, 181, 619, 929, 3407, 10949, 13241, 13873, 16519, 201359, 396413, 1888279, 3300593, ..., 4939471, ..., 5154509, ... (sequence A004061 in the OEIS).

The first few base-6 repunit primes are

7, 43, 55987, 7369130657357778596659, 3546245297457217493590449191748546458005595187661976371, 133733063818254349335501779590081460423013416258060407531857720755181857441961908284738707408499507 (sequence A165210 in the OEIS),

corresponding to ${\displaystyle n}$of

2, 3, 7, 29, 71, 127, 271, 509, 1049, 6389, 6883, 10613, 19889, 79987, 608099, 1365019, 3360347, ... (sequence A004062 in the OEIS).

The first few base-7 repunit primes are

2801, 16148168401, 85053461164796801949539541639542805770666392330682673302530819774105141531698707146930307290253537320447270457,
138502212710103408700774381033135503926663324993317631729227790657325163310341833227775945426052637092067324133850503035623601

corresponding to ${\displaystyle n}$of

5, 13, 131, 149, 1699, 14221, 35201, 126037, 371669, 1264699, ... (sequence A004063 in the OEIS).

The only base-8 repunit prime is 73 (${\displaystyle 111_{8}}$). ${\displaystyle 8^{n}-1=\left(4^{n}+2^{n}+1\right)\left(2^{n}-1\right)}$, and 7 always divides ${\displaystyle 4^{n}+2^{n}+1}$when n is not divisible by 3 and ${\displaystyle 2^{n}-1}$when n is divisible by 3. For n greater than 3, both ${\displaystyle 4^{n}+2^{n}+1}$and ${\displaystyle 2^{n}-1}$are greater than 7, so removing the factor of 7 still leaves two factors greater than 1. Therefore, the number cannot be prime.

There are no base-9 repunit primes. ${\displaystyle 9^{n}-1=\left(3^{n}+1\right)\left(3^{n}-1\right)}$, and ${\displaystyle 3^{n}+1}$and ${\displaystyle 3^{n}-1}$are even, and one of ${\displaystyle 3^{n}+1}$and ${\displaystyle 3^{n}-1}$is divisible by 4. For n greater than 1, both ${\displaystyle 3^{n}+1}$and ${\displaystyle 3^{n}-1}$are greater than 4, so removing the factor of 8 (which is equivalent to removing the factor 4 from ${\displaystyle 3^{n}+1}$or ${\displaystyle 3^{n}-1}$, and removing the factor 2 from the other number) still leaves two factors greater than 1. Therefore, the number cannot be prime.

The first few base-11 repunit primes are

50544702849929377, 6115909044841454629, 1051153199500053598403188407217590190707671147285551702341089650185945215953, 567000232521795739625828281267171344486805385881217575081149660163046217465544573355710592079769932651989153833612198334843467861091902034340949

corresponding to ${\displaystyle n}$of

17, 19, 73, 139, 907, 1907, 2029, 4801, 5153, 10867, 20161, 293831, 1868983, ... (sequence A005808 in the OEIS).

The first few base-12 repunit primes are

13, 157, 22621, 29043636306420266077, 43570062353753446053455610056679740005056966111842089407838902783209959981593077811330507328327968191581, 388475052482842970801320278964160171426121951256610654799120070705613530182445862582590623785872890159937874339918941

corresponding to ${\displaystyle n}$of

2, 3, 5, 19, 97, 109, 317, 353, 701, 9739, 14951, 37573, 46889, 769543, ... (sequence A004064 in the OEIS).

The only base-16 repunit prime is 17 (${\displaystyle 11_{16}}$). ${\displaystyle 16^{n}-1=\left(4^{n}+1\right)\left(4^{n}-1\right)}$, and 3 always divides ${\displaystyle 4^{n}-1}$, and 5 always divides ${\displaystyle 4^{n}+1}$when n is odd and ${\displaystyle 4^{n}-1}$when n is even. For n greater than 2, both ${\displaystyle 4^{n}+1}$and ${\displaystyle 4^{n}-1}$are greater than 15, so removing the factor of 15 still leaves two factors greater than 1. Therefore, the number cannot be prime.

The first few base-20 repunit primes are

421, 10778947368421, 689852631578947368421

corresponding to ${\displaystyle n}$of

3, 11, 17, 1487, 31013, 48859, 61403, 472709, 984349, ... (sequence A127995 in the OEIS).

Smallest base ${\displaystyle b}$such that ${\displaystyle R_{p}(b)}$is prime (where ${\displaystyle p}$is the ${\displaystyle n}$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, ... (sequence A066180 in the OEIS)

Smallest base ${\displaystyle b}$such that ${\displaystyle R_{p}(-b)}$is prime (where ${\displaystyle p}$is the ${\displaystyle n}$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, ... (sequence A103795 in the OEIS)

Smallest prime ${\displaystyle p>2}$such that ${\displaystyle R_{p}(b)}$is prime are (start with ${\displaystyle b=2}$, 0 if no such ${\displaystyle p}$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, ... (sequence A128164 in the OEIS)

Smallest prime ${\displaystyle p>2}$such that ${\displaystyle R_{p}(-b)}$is prime are (start with ${\displaystyle b=2}$, 0 if no such ${\displaystyle p}$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, (>800000), 19, 7, 3, 7, ... (sequence A084742 in the OEIS)

^* 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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