Williams number

In number theory, a Williams number base b is a natural number of the form $(b-1)\cdot b^{n}-1$ 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.

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	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, ...	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, ...	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, ...	A056725
11	1, 3, 37, 119, 255, 355, 371, 497, 1759, 34863, 50719, 147709, ...	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, ...	A297348
14	1, 3, 5, 27, 35, 165, 209, 2351, 11277, ...	A273523
15	14, 33, 43, ...
16	1, 20, 29, 43, 56, 251, ...
17	1, 3, 71, 139, 265, 793, 1729, ...
18	2, 6, 26, 79, 91, 96, 416, 554, 1910, ...
19	6, 9, 20, 43, 174, 273, 428, 1388, ...
20	1, 219, 223, ...
21	1, 2, 7, 24, 31, 60, 230, 307, 750, 1131, 1665, 1827, ...
22	1, 2, 5, 19, 141, 302, 337, ...
23	55, 103, 115, 131, 535, 1183, ...
24	12, 18, 63, 153, 221, 1256, ...
25	1, 5, 7, 30, 75, 371, 383, 609, 819, 855, ...
26	133, 205, 215, 1649, ...
27	1, 3, 5, 13, 15, 31, 55, 151, 259, 479, 734, 1775, 2078, ...
28	20, 1091, ...
29	1, 7, 11, 57, 69, 235, ...
30	2, 83, 566, 938, 1934, 2323, 3032, ...

As of September 2018, the largest known Williams prime base 3 is 2×3^1360104−1^[2].

Generalization

A Williams number of the second kind base b is a natural number of the form $(b-1)\cdot b^{n}+1$ 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	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, ...
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, ...
14	2, 40, 402, 1070, ...
15	1, 3, 4, 9, 11, 14, 23, 122, 141, 591, 2115, 2398, 2783, ...
16	1, 3, 11, 12, 28, 42, 225, 702, 782, 972, 1701, 1848, ...
17	4, 20, 320, 736, 2388, ...
18	1, 6, 9, 12, 22, 30, 102, 154, 600, ...
19	29, 32, 59, 65, 303, 1697, ...
20	14, 18, 20, 38, 108, 150, 640, ...
21	1, 2, 3, 4, 12, 17, 38, 54, 56, 123, 165, 876, 1110, 1178, 2465, ...
22	1, 9, 53, 261, 1491, 2120, 2592, ...
23	14, 62, 84, ...
24	2, 4, 9, 42, 47, 54, 89, 102, 118, 269, 273, 316, 698, 1872, 2126, ...
25	1, 4, 162, 1359, 2620, ...
26	2, 18, 100, 1178, 1196, ...
27	4, 5, 167, 408, 416, 701, 707, 1811, 3268, ...
28	1, 2, 136, 154, 524, 1234, 2150, 2368, ...
29	2, 4, 6, 44, 334, ...
30	4, 5, 9, 18, 71, 124, 165, 172, 888, 2218, ...

As of September 2018, the largest known Williams prime of the second kind base 3 is 2×3^1175232+1^[3].

A Williams number of the third kind base b is a natural number of the form $(b+1)\cdot b^{n}-1$ 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 $(b+1)\cdot b^{n}+1$ 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 $b\equiv 1{\bmod {3}}$ .

b	numbers n such that $(b+1)\cdot b^{n}-1$ is prime	numbers n such that $(b+1)\cdot b^{n}+1$ is prime
2	A002235	A002253
3	A005540	A005537
5	A257790	A143279
10	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 chose the numerator of the numbers, then we get these numbers:

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

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

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

Dual Williams numbers of the fourth kind base b: numbers of the form $b^{n}+(b+1)$ 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 $b^{n}-(b-1)$ is (probable) prime (dual Williams primes of the first kind)	numbers n such that $b^{n}+(b-1)$ is (probable) prime (dual Williams primes of the second kind)	numbers n such that $b^{n}-(b+1)$ is (probable) prime (dual Williams primes of the third kind)	numbers n such that $b^{n}+(b+1)$ is (probable) prime (dual Williams primes of the fourth kind)
2	A000043	(see Fermat prime)	A050414	A057732
3	A014224	A051783	A058959	A058958
4	A059266	A089437	A217348	(not exist)
5	A059613	A124621	A165701	A089142
6	A059614	A145106	A217352	A217351
7	A191469	A217130	A217131	(not exist)
8	A217380	A217381	A217383	A217382
9	A177093	A217385	A217493	A217492
10	A095714	A088275	A092767	(not exist)

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

References

External links

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[1] Williams primes

[2] The Prime Database: 2·3^1360104 − 1

[3] The Prime Database: 2·3^1175232 + 1

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
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