Williams number

Not to be confused with Newman–Shanks–Williams prime.

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]

It is conjectured that for every b ≥ 2, there are infinitely many Williams primes for base b.

See also

• Thabit number