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. 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.

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

• Thabit number

• The primality of certain integers of the form 2Ar − 1
• Some prime numbers of the forms 2·3 + 1 and 2·3 − 1
• Williams prime at PrimeWiki