In number theory, a Williams number base b is a natural number of the form 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 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 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 .

b numbers n such that is prime numbers n such that is prime
2 OEIS: A002235 OEIS: A002253
3 OEIS: A005540 OEIS: A005537
5 OEIS: A257790 OEIS: A143279
10 OEIS: A111391 (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 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. (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 is (probable) prime (dual Williams primes of the first kind) numbers n such that is (probable) prime (dual Williams primes of the second kind) numbers n such that is (probable) prime (dual Williams primes of the third kind) numbers n such that is (probable) prime (dual Williams primes of the fourth kind)
2 OEIS: A000043 (see Fermat prime) OEIS: A050414 OEIS: A057732
3 OEIS: A014224 OEIS: A051783 OEIS: A058959 OEIS: A058958
4 OEIS: A059266 OEIS: A089437 OEIS: A217348 (not exist)
5 OEIS: A059613 OEIS: A124621 OEIS: A165701 OEIS: A089142
6 OEIS: A059614 OEIS: A145106 OEIS: A217352 OEIS: A217351
7 OEIS: A191469 OEIS: A217130 OEIS: A217131 (not exist)
8 OEIS: A217380 OEIS: A217381 OEIS: A217383 OEIS: A217382
9 OEIS: A177093 OEIS: A217385 OEIS: A217493 OEIS: A217492
10 OEIS: A095714 OEIS: A088275 OEIS: A092767 (not exist)

(for the smallest dual Williams primes of the 1st, 2nd and 3rd kinds base b, see OEIS: A113516, OEIS: A076845 and OEIS: A178250)

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.

See also

  • Thabit number, which is exactly the Williams number of the third kind base 2

References

  1. Williams primes
  2. See Table 1 in the last page of the paper: Williams, H. C. (1981). "The primality of certain integers of the form 2 A rn – 1". Acta Arith. 39: 7–17. doi:10.4064/aa-39-1-7-17.
  3. The Prime Database: 2·31360104  1
  4. The Prime Database: 2·31175232 + 1
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