Williams number explained

Williams number should not be confused with Newman–Shanks–Williams prime.

In number theory, a Williams number base b is a natural number of the form

(b -1 ) ⋅ 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

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

Notes and References

1. http://harvey563.tripod.com/wills.txt Williams primes
2. See Table 1 in the last page of the paper: Williams . H. C. . Hugh C. Williams . The primality of certain integers of the form 2 A rⁿ – 1 . . 39 . 1981 . 7–17 . 10.4064/aa-39-1-7-17 . free .