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.

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, ...

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

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)

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

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	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.

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