# Williams number

In number theory, a Williams number base b is a natural number of the form ${\displaystyle (b-1)\cdot b^{n}-1}$ for integers b ≥ 2 and n ≥ 1.[1] The Williams numbers base 2 are exactly the Mersenne numbers.

## Williams prime

A Williams prime is a Williams number that is prime. They were considered by Hugh C. Williams.[2]

Least n ≥ 1 such that (b−1)·bn − 1 is prime are: (start with b = 2)

2, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 2, 1, 14, 1, 1, 2, 6, 1, 1, 1, 55, 12, 1, 133, 1, 20, 1, 2, 1, 1, 2, 15, 3, 1, 7, 136211, 1, 1, 7, 1, 7, 7, 1, 1, 1, 2, 1, 25, 1, 5, 3, 1, 1, 1, 1, 2, 3, 1, 1, 899, 3, 11, 1, 1, 1, 63, 1, 13, 1, 25, 8, 3, 2, 7, 1, 44, 2, 11, 3, 81, 21495, 1, 2, 1, 1, 3, 25, 1, 519, 77, 476, 1, 1, 2, 1, 4983, 2, 2, ...
 b numbers n ≥ 1 such that (b−1)×bn−1 is prime (these n are checked up to 25000) OEIS sequence 2 2, 3, 5, 7, 13, 17, 19, 31, 61, 89, 107, 127, 521, 607, 1279, 2203, 2281, 3217, 4253, 4423, 9689, 9941, 11213, 19937, 21701, 23209, 44497, 86243, 110503, 132049, 216091, 756839, 859433, 1257787, 1398269, 2976221, 3021377, 6972593, 13466917, 20996011, 24036583, 25964951, 30402457, 32582657, 37156667, 42643801, 43112609, 57885161, 74207281, 77232917, 82589933, ... A000043 3 1, 2, 3, 7, 8, 12, 20, 23, 27, 35, 56, 62, 68, 131, 222, 384, 387, 579, 644, 1772, 3751, 5270, 6335, 8544, 9204, 12312, 18806, 21114, 49340, 75551, 90012, 128295, 143552, 147488, 1010743, 1063844, 1360104, ... A003307 4 1, 2, 3, 9, 17, 19, 32, 38, 47, 103, 108, 153, 162, 229, 235, 637, 1638, 2102, 2567, 6338, 7449, 12845, 20814, 40165, 61815, 77965, 117380, 207420, 351019, 496350, 600523, 1156367, 2117707, 5742009, 5865925, 5947859, ... A272057 5 1, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751, ... A046865 6 1, 2, 6, 7, 11, 23, 33, 48, 68, 79, 116, 151, 205, 1016, 1332, 1448, 3481, 3566, 3665, 11233, 13363, 29166, 44358, 58530, 191706, ... A079906 7 1, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326, ... A046866 8 3, 7, 15, 59, 6127, 8703, 11619, 23403, 124299, ... A268061 9 1, 2, 5, 25, 85, 92, 97, 649, 2017, 2978, 3577, 4985, 17978, 21365, 66002, 95305, 142199, ... A268356 10 1, 3, 7, 19, 29, 37, 93, 935, 8415, 9631, 11143, 41475, 41917, 48051, 107663, 212903, 223871, 260253, 364521, 383643, 1009567, ... A056725 11 1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709, 263893, ... A046867 12 1, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961, ... A079907 13 2, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466, ... A297348 14 1, 3, 5, 27, 35, 165, 209, 2351, 11277, 21807, 25453, 52443, ... A273523 15 14, 33, 43, 20885, ... 16 1, 20, 29, 43, 56, 251, 25985, 27031, 142195, 164066, ... 17 1, 3, 71, 139, 265, 793, 1729, 18069, ... 18 2, 6, 26, 79, 91, 96, 416, 554, 1910, 4968, ... 19 6, 9, 20, 43, 174, 273, 428, 1388, ... 20 1, 219, 223, 3659, ... 21 1, 2, 7, 24, 31, 60, 230, 307, 750, 1131, 1665, 1827, 8673, ... 22 1, 2, 5, 19, 141, 302, 337, 4746, 5759, 16530, ... 23 55, 103, 115, 131, 535, 1183, 9683, ... 24 12, 18, 63, 153, 221, 1256, 13116, 15593, ... 25 1, 5, 7, 30, 75, 371, 383, 609, 819, 855, 7130, 7827, 9368, ... 26 133, 205, 215, 1649, ... 27 1, 3, 5, 13, 15, 31, 55, 151, 259, 479, 734, 1775, 2078, 6159, 6393, 9013, ... 28 20, 1091, 5747, 6770, ... 29 1, 7, 11, 57, 69, 235, 16487, ... 30 2, 83, 566, 938, 1934, 2323, 3032, 7889, 8353, 9899, 11785, ...

As of September 2018, the largest known Williams prime base 3 is 2×31360104−1.[3]

## Generalization

A Williams number of the second kind base b is a natural number of the form ${\displaystyle (b-1)\cdot b^{n}+1}$ for integers b ≥ 2 and n ≥ 1, a Williams prime of the second kind is a Williams number of the second kind that is prime. The Williams primes of the second kind base 2 are exactly the Fermat primes.

Least n ≥ 1 such that (b−1)·bn + 1 is prime are: (start with b = 2)

1, 1, 1, 2, 1, 1, 2, 1, 3, 10, 3, 1, 2, 1, 1, 4, 1, 29, 14, 1, 1, 14, 2, 1, 2, 4, 1, 2, 4, 5, 12, 2, 1, 2, 2, 9, 16, 1, 2, 80, 1, 2, 4, 2, 3, 16, 2, 2, 2, 1, 15, 960, 15, 1, 4, 3, 1, 14, 1, 6, 20, 1, 3, 946, 6, 1, 18, 10, 1, 4, 1, 5, 42, 4, 1, 828, 1, 1, 2, 1, 12, 2, 6, 4, 30, 3, 3022, 2, 1, 1, 8, 2, 4, 4, 2, 11, 8, 2, 1, ... (sequence A305531 in the OEIS)
 b numbers n ≥ 1 such that (b−1)×bn+1 is prime (these n are checked up to 25000) OEIS sequence 2 1, 2, 4, 8, 16, ... 3 1, 2, 4, 5, 6, 9, 16, 17, 30, 54, 57, 60, 65, 132, 180, 320, 696, 782, 822, 897, 1252, 1454, 4217, 5480, 6225, 7842, 12096, 13782, 17720, 43956, 64822, 82780, 105106, 152529, 165896, 191814, 529680, 1074726, 1086112, 1175232, ... A003306 4 1, 3, 4, 6, 9, 15, 18, 33, 138, 204, 219, 267, 1104, 1408, 1584, 1956, 17175, 21147, 24075, 27396, 27591, 40095, 354984, 400989, 916248, 1145805, 2541153, 5414673, ... A326655 5 2, 6, 18, 50, 290, 2582, 20462, 23870, 26342, 31938, 38122, 65034, 70130, 245538, ... A204322 6 1, 2, 4, 17, 136, 147, 203, 590, 754, 964, 970, 1847, 2031, 2727, 2871, 5442, 7035, 7266, 11230, 23307, 27795, 34152, 42614, 127206, 133086, ... A247260 7 1, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572, ... A245241 8 2, 40, 58, 60, 130, 144, 752, 7462, 18162, 69028, 187272, 268178, 270410, 497284, 713304, 722600, 1005254, ... A269544 9 1, 4, 5, 11, 26, 29, 38, 65, 166, 490, 641, 2300, 9440, 44741, 65296, 161930, ... A056799 10 3, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240, ... A056797 11 10, 24, 864, 2440, 9438, 68272, 148602, ... A057462 12 3, 4, 35, 119, 476, 507, 6471, 13319, 31799, ... A251259 13 1, 2, 4, 21, 34, 48, 53, 160, 198, 417, 773, 1220, 5361, 6138, 15557, 18098, ... 14 2, 40, 402, 1070, 6840, ... 15 1, 3, 4, 9, 11, 14, 23, 122, 141, 591, 2115, 2398, 2783, 3692, 3748, 10996, 16504, ... 16 1, 3, 11, 12, 28, 42, 225, 702, 782, 972, 1701, 1848, 8556, 8565, 10847, 12111, 75122, 183600, 307400, 342107, 416936, ... 17 4, 20, 320, 736, 2388, 3344, 8140, ... 18 1, 6, 9, 12, 22, 30, 102, 154, 600, ... 19 29, 32, 59, 65, 303, 1697, 5358, 9048, ... 20 14, 18, 20, 38, 108, 150, 640, 8244, ... 21 1, 2, 3, 4, 12, 17, 38, 54, 56, 123, 165, 876, 1110, 1178, 2465, 3738, 7092, 8756, 15537, 19254, 24712, ... 22 1, 9, 53, 261, 1491, 2120, 2592, 6665, 9460, 15412, 24449, ... 23 14, 62, 84, 8322, 9396, 10496, 24936, ... 24 2, 4, 9, 42, 47, 54, 89, 102, 118, 269, 273, 316, 698, 1872, 2126, 22272, ... 25 1, 4, 162, 1359, 2620, ... 26 2, 18, 100, 1178, 1196, 16644, ... 27 4, 5, 167, 408, 416, 701, 707, 1811, 3268, 3508, 7020, 7623, 16449, ... 28 1, 2, 136, 154, 524, 1234, 2150, 2368, 7222, 10082, 14510, 16928, ... 29 2, 4, 6, 44, 334, 24714, ... 30 4, 5, 9, 18, 71, 124, 165, 172, 888, 2218, 3852, 17871, 23262, ...

As of September 2018, the largest known Williams prime of the second kind base 3 is 2×31175232+1.[4]

A Williams number of the third kind base b is a natural number of the form ${\displaystyle (b+1)\cdot b^{n}-1}$ for integers b ≥ 2 and n ≥ 1, the Williams number of the third kind base 2 are exactly the Thabit numbers. A Williams prime of the third kind is a Williams number of the third kind that is prime.

A Williams number of the fourth kind base b is a natural number of the form ${\displaystyle (b+1)\cdot b^{n}+1}$ for integers b ≥ 2 and n ≥ 1, a Williams prime of the fourth kind is a Williams number of the fourth kind that is prime, such primes do not exist for ${\displaystyle b\equiv 1{\bmod {3}}}$.

 b numbers n such that ${\displaystyle (b+1)\cdot b^{n}-1}$ is prime numbers n such that ${\displaystyle (b+1)\cdot b^{n}+1}$ is prime 2 3 5 10 (not exist)

It is conjectured that for every b ≥ 2, there are infinitely many Williams primes of the first kind (the original Williams primes) base b, infinitely many Williams primes of the second kind base b, and infinitely many Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many Williams primes of the fourth kind base b.

## Dual form

If we let n take negative values, and choose the numerator of the numbers, then we get these numbers:

Dual Williams numbers of the first kind base b: numbers of the form ${\displaystyle b^{n}-(b-1)}$ with b ≥ 2 and n ≥ 1.

Dual Williams numbers of the second kind base b: numbers of the form ${\displaystyle b^{n}+(b-1)}$ with b ≥ 2 and n ≥ 1.

Dual Williams numbers of the third kind base b: numbers of the form ${\displaystyle b^{n}-(b+1)}$ with b ≥ 2 and n ≥ 1.

Dual Williams numbers of the fourth kind base b: numbers of the form ${\displaystyle b^{n}+(b+1)}$ with b ≥ 2 and n ≥ 1. (not exist when b = 1 mod 3)

Unlike the original Williams primes of each kind, some large dual Williams primes of each kind are only probable primes, since for these primes N, neither N−1 not N+1 can be trivially written into a product.

 b numbers n such that ${\displaystyle b^{n}-(b-1)}$ is (probable) prime (dual Williams primes of the first kind) numbers n such that ${\displaystyle b^{n}+(b-1)}$ is (probable) prime (dual Williams primes of the second kind) numbers n such that ${\displaystyle b^{n}-(b+1)}$ is (probable) prime (dual Williams primes of the third kind) numbers n such that ${\displaystyle b^{n}+(b+1)}$ is (probable) prime (dual Williams primes of the fourth kind) 2 (see Fermat prime) 3 4 (not exist) 5 6 7 (not exist) 8 9 10 (not exist)

(for the smallest dual Williams primes of the 1st, 2nd and 3rd kinds base b, see , and )

It is conjectured that for every b ≥ 2, there are infinitely many dual Williams primes of the first kind (the original Williams primes) base b, infinitely many dual Williams primes of the second kind base b, and infinitely many dual Williams primes of the third kind base b. Besides, if b is not = 1 mod 3, then there are infinitely many dual Williams primes of the fourth kind base b.