Difference between revisions of "Williams prime"

Revision as of 12:42, 15 May 2019

A Williams number is a natural number of the form (b-1)•bⁿ-1 for integers b ≥ 2 and n ≥ 1.

A Williams prime is a Williams number which is prime.

Varying both signs, there're four different types of numbers similiar to Williams numbers.

Lists of primes for bases b and n-values can be found here:

Type	Category	List table ^[1]	List smallest ^[2]
MM: `(b-1)`•`b`ⁿ-1	here	here	here
MP: `(b-1)`•`b`ⁿ+1	here	here	here
PM: `(b+1)`•`b`ⁿ-1	here	here	here
PP: `(b+1)`•`b`ⁿ+1	here	here	here ^[3]

↑ The list table of any type contains only bases which are included as a separate page.
↑ The list of smallest primes of any base is an ASCII file for 2 ≤ b ≤ 1024. Any unknown value is given with the search range in brackets or empty.
↑ Values for bases b == 1 mod 3 are always divisible by 3, so not listed here.

H. C. Williams: "The primality of certain integers of the form 2Ar^n-1", Acta Arith. 39 (1981), 7-17
A. Stein, H. C. Williams: "Explicit primality criteria for (p−1)pⁿ−1", Math. Comp. 69 (2000), 1721-1734
Steven Harvey: Search for original Williams primes
Mauro Fiorentini: Type MM, Type MP, Type PM, Type PP for 0 ≥ n ≥ 1000 (mostly) and 1 ≥ b ≥ 1000 (2016)
Williams number

Number classes

General numbers

Special numbers

Prime numbers

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 *A. Stein, H. C. Williams: [https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf "Explicit primality criteria for (p−1)p<sup>n</sup>−1"], Math. Comp. 69 (2000), 1721-1734
 *Steven Harvey: [http://harvey563.tripod.com/wills.txt Search for original Williams primes]
+*Mauro Fiorentini: [http://www.bitman.name/math/table/484 Type MM], [http://www.bitman.name/math/table/477 Type MP], [http://www.bitman.name/math/table/471 Type PM], [http://www.bitman.name/math/table/474 Type PP] for 0 ≥ ''n'' ≥ 1000 (mostly) and 1 ≥ ''b'' ≥ 1000 (2016)
 *[[Wikipedia:Williams number|Williams number]]
 {{Navbox NumberClasses}}
 [[Category:Williams prime| ]]