Williams number#Williams prime

{{Short description|Class of numbers in number theory}}

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{{distinguish|Newman–Shanks–Williams prime}}

In number theory, a Williams number base b is a natural number of the form $(b-1)\; \backslash cdot\; b^n-1$ for integers b ≥ 2 and n ≥ 1.[http://harvey563.tripod.com/wills.txt Williams primes] 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.See Table 1 in the last page of the paper: {{cite journal |last=Williams |first=H. C. |author-link=Hugh C. Williams |title= The primality of certain integers of the form 2 A rⁿ – 1 |journal=Acta Arith. |volume=39 |date=1981 |pages=7–17 |doi=10.4064/aa-39-1-7-17 |doi-access=free }}

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