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 $(b-1)\cdot b^n-1$ 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.

See also

• Thabit number

External links

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