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Difference between revisions of "Williams prime"
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*H. C. Williams: [http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf "The primality of certain integers of the form 2Ar^n-1"], Acta Arith. 39 (1981), 7-17 | *H. C. Williams: [http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf "The primality of certain integers of the form 2Ar^n-1"], Acta Arith. 39 (1981), 7-17 | ||
*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 | *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 | + | *Steven Harvey: [http://harvey563.tripod.com/wills.txt Search for Williams primes]: only Type MM {{Kbn|(b-1)|b|n}} for 2 ≤ ''b'' ≤ 2048, 1 ≤ ''n'' ≤ 512 and some higher (2006-2019) |
| − | *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 | + | *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) |
| + | *Eric Chen: [https://www.mersenneforum.org/showthread.php?t=21818 Thread] at [[MersenneForum]] including dual forms for 0 ≤ ''n'' ≤ 5000 and 1 ≤ ''b'' ≤ 64 and some higher (2016-2019) | ||
*[[Wikipedia:Williams number|Williams number]] | *[[Wikipedia:Williams number|Williams number]] | ||
{{Navbox NumberClasses}} | {{Navbox NumberClasses}} | ||
[[Category:Williams prime| ]] | [[Category:Williams prime| ]] | ||
Revision as of 07:42, 22 May 2019
Definition
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.
Generalization
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)•bn-1 | here | here | here |
| MP: (b-1)•bn+1 | here | here | here |
| PM: (b+1)•bn-1 | here | here | here |
| PP: (b+1)•bn+1 | here | here | here [3] |
Notes
- ↑ 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.
External links
- 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 Williams primes: only Type MM (b-1)•bn-1 for 2 ≤ b ≤ 2048, 1 ≤ n ≤ 512 and some higher (2006-2019)
- Mauro Fiorentini: Type MM, Type MP, Type PM, Type PP for 0 ≤ n ≤ 1000 (mostly) and 1 ≤ b ≤ 1000 (2016)
- Eric Chen: Thread at MersenneForum including dual forms for 0 ≤ n ≤ 5000 and 1 ≤ b ≤ 64 and some higher (2016-2019)
- Williams number
Number classes
| General numbers |
| Special numbers |
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| Prime numbers |
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