Williams number


In number theory, a Williams number base b is a natural number of the form for integers b ≥ 2 and n ≥ 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.
Least n ≥ 1 such that ·bn − 1 is prime are:
bnumbers n ≥ 1 such that ×bn−1 is prime OEIS sequence
22, 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,...
31, 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,...
41, 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,...
51, 3, 9, 13, 15, 25, 39, 69, 165, 171, 209, 339, 2033, 6583, 15393, 282989, 498483, 504221, 754611, 864751,...
61, 2, 6, 7, 11, 23, 33, 48, 68, 79, 116, 151, 205, 1016, 1332, 1448, 3481, 3566, 3665, 11233, 13363, 29166, 44358, 58530, 191706,...
71, 2, 7, 18, 55, 69, 87, 119, 141, 189, 249, 354, 1586, 2135, 2865, 2930, 4214, 7167, 67485, 74402, 79326,...
83, 7, 15, 59, 6127, 8703, 11619, 23403, 124299,...
91, 2, 5, 25, 85, 92, 97, 649, 2017, 2978, 3577, 4985, 17978, 21365, 66002, 95305, 142199,...
101, 3, 7, 19, 29, 37, 93, 935, 8415, 9631, 11143, 41475, 41917, 48051, 107663, 212903, 223871, 260253, 364521, 383643, 1009567,...
111, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709, 263893,...
121, 2, 21, 25, 33, 54, 78, 235, 1566, 2273, 2310, 4121, 7775, 42249, 105974, 138961,...
132, 7, 11, 36, 164, 216, 302, 311, 455, 738, 1107, 2244, 3326, 4878, 8067, 46466,...
141, 3, 5, 27, 35, 165, 209, 2351, 11277, 21807, 25453, 52443,...
1514, 33, 43, 20885,...
161, 20, 29, 43, 56, 251, 25985, 27031, 142195, 164066,...
171, 3, 71, 139, 265, 793, 1729, 18069,...
182, 6, 26, 79, 91, 96, 416, 554, 1910, 4968,...
196, 9, 20, 43, 174, 273, 428, 1388,...
201, 219, 223, 3659,...
211, 2, 7, 24, 31, 60, 230, 307, 750, 1131, 1665, 1827, 8673,...
221, 2, 5, 19, 141, 302, 337, 4746, 5759, 16530,...
2355, 103, 115, 131, 535, 1183, 9683,...
2412, 18, 63, 153, 221, 1256, 13116, 15593,...
251, 5, 7, 30, 75, 371, 383, 609, 819, 855, 7130, 7827, 9368,...
26133, 205, 215, 1649,...
271, 3, 5, 13, 15, 31, 55, 151, 259, 479, 734, 1775, 2078, 6159, 6393, 9013,...
2820, 1091, 5747, 6770,...
291, 7, 11, 57, 69, 235, 16487,...
302, 83, 566, 938, 1934, 2323, 3032, 7889, 8353, 9899, 11785,...

, the largest known Williams prime base 3 is 2×31360104−1.

Generalization

A Williams number of the second kind base b is a natural number of the form 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 ·bn + 1 is prime are:
bnumbers n ≥ 1 such that ×bn+1 is prime OEIS sequence
21, 2, 4, 8, 16,...
31, 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,...
41, 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,...
52, 6, 18, 50, 290, 2582, 20462, 23870, 26342, 31938, 38122, 65034, 70130, 245538,...
61, 2, 4, 17, 136, 147, 203, 590, 754, 964, 970, 1847, 2031, 2727, 2871, 5442, 7035, 7266, 11230, 23307, 27795, 34152, 42614, 127206, 133086,...
71, 4, 9, 99, 412, 2633, 5093, 5632, 28233, 36780, 47084, 53572,...
82, 40, 58, 60, 130, 144, 752, 7462, 18162, 69028, 187272, 268178, 270410, 497284, 713304, 722600, 1005254,...
91, 4, 5, 11, 26, 29, 38, 65, 166, 490, 641, 2300, 9440, 44741, 65296, 161930,...
103, 4, 5, 9, 22, 27, 36, 57, 62, 78, 201, 537, 696, 790, 905, 1038, 66886, 70500, 91836, 100613, 127240,...
1110, 24, 864, 2440, 9438, 68272, 148602,...
123, 4, 35, 119, 476, 507, 6471, 13319, 31799,...
131, 2, 4, 21, 34, 48, 53, 160, 198, 417, 773, 1220, 5361, 6138, 15557, 18098,...
142, 40, 402, 1070, 6840,...
151, 3, 4, 9, 11, 14, 23, 122, 141, 591, 2115, 2398, 2783, 3692, 3748, 10996, 16504,...
161, 3, 11, 12, 28, 42, 225, 702, 782, 972, 1701, 1848, 8556, 8565, 10847, 12111, 75122, 183600, 307400, 342107, 416936,...
174, 20, 320, 736, 2388, 3344, 8140,...
181, 6, 9, 12, 22, 30, 102, 154, 600,...
1929, 32, 59, 65, 303, 1697, 5358, 9048,...
2014, 18, 20, 38, 108, 150, 640, 8244,...
211, 2, 3, 4, 12, 17, 38, 54, 56, 123, 165, 876, 1110, 1178, 2465, 3738, 7092, 8756, 15537, 19254, 24712,...
221, 9, 53, 261, 1491, 2120, 2592, 6665, 9460, 15412, 24449,...
2314, 62, 84, 8322, 9396, 10496, 24936,...
242, 4, 9, 42, 47, 54, 89, 102, 118, 269, 273, 316, 698, 1872, 2126, 22272,...
251, 4, 162, 1359, 2620,...
262, 18, 100, 1178, 1196, 16644,...
274, 5, 167, 408, 416, 701, 707, 1811, 3268, 3508, 7020, 7623, 16449,...
281, 2, 136, 154, 524, 1234, 2150, 2368, 7222, 10082, 14510, 16928,...
292, 4, 6, 44, 334, 24714,...
304, 5, 9, 18, 71, 124, 165, 172, 888, 2218, 3852, 17871, 23262,...

, the largest known Williams prime of the second kind base 3 is 2×31175232+1.
A Williams number of the third kind base b is a natural number of the form 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 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.
bnumbers n such that is primenumbers n such that is prime
2
3
5
10

It is conjectured that for every b ≥ 2, there are infinitely many Williams primes of the first kind 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 with b ≥ 2 and n ≥ 1.
Dual Williams numbers of the second kind base b: numbers of the form with b ≥ 2 and n ≥ 1.
Dual Williams numbers of the third kind base b: numbers of the form with b ≥ 2 and n ≥ 1.
Dual Williams numbers of the fourth kind base b: numbers of the form with b ≥ 2 and n ≥ 1.
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.
bnumbers n such that is prime numbers n such that is prime numbers n such that is prime numbers n such that is prime
2
3
4
5
6
7
8
9
10

It is conjectured that for every b ≥ 2, there are infinitely many dual Williams primes of the first kind 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.