Williams number
In number theory, a Williams number base b is a natural number of the form ( b − 1 ) ⋅ b n − 1 {\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.
In number theory, a Williams number base b is a natural number of the form 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
[edit]References
[edit]- ^ Williams primes
- ^ See Table 1 in the last page of the paper: Williams, H. C. (1981). "The primality of certain integers of the form 2 A rn – 1". Acta Arith. 39: 7–17. doi:10.4064/aa-39-1-7-17.
External links
[edit]- 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
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