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Difference between revisions of "Williams prime"
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==Definition== | ==Definition== | ||
| − | A '''Williams number''' is a [[natural number]] of the form {{Kbn|(b-1)|b|n}} for integers ''b | + | A '''Williams number''' is a [[natural number]] of the form {{Kbn|(b-1)|b|n}} for integers ''b'' ≥ 2 and ''n'' ≥ 1. |
A '''Williams prime''' is a Williams number which is [[prime]]. | A '''Williams prime''' is a Williams number which is [[prime]]. | ||
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Lists of primes for bases ''b'' and ''n''-values can be found here: | Lists of primes for bases ''b'' and ''n''-values can be found here: | ||
{| class="wikitable" | {| class="wikitable" | ||
| − | ! Type !! Category !! | + | ! Type !! Category !! Table <ref>The table contains only bases which are included as a separate page.</ref> !! Smallest <ref>The list of smallest primes of any base is an ASCII file for 2 ≤ ''b'' ≤ 1024. For unknown values only the base is given.</ref> |
|- | |- | ||
| − | | MM: {{Kbn|(b-1)|b|n}} || [[:Category:Williams prime MM|here]] ||[[Williams prime MM table|here]] || [[Williams prime MM least|here]] | + | | MM: {{Kbn|(b-1)|b|n}} || [[:Category:Williams prime MM|here]] ||[[Williams prime MM table|here]] || [[Williams prime MM least|here]]<ref>The list contains values for 2 ≤ ''b'' ≤ 2049.</ref> |
|- | |- | ||
| MP: {{Kbn|+|(b-1)|b|n}} || [[:Category:Williams prime MP|here]] ||[[Williams prime MP table|here]] || [[Williams prime MP least|here]] | | MP: {{Kbn|+|(b-1)|b|n}} || [[:Category:Williams prime MP|here]] ||[[Williams prime MP table|here]] || [[Williams prime MP least|here]] | ||
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| PM: {{Kbn|(b+1)|b|n}} || [[:Category:Williams prime PM|here]] ||[[Williams prime PM table|here]] || [[Williams prime PM least|here]] | | PM: {{Kbn|(b+1)|b|n}} || [[:Category:Williams prime PM|here]] ||[[Williams prime PM table|here]] || [[Williams prime PM least|here]] | ||
|- | |- | ||
| − | | PP: {{Kbn|+|(b+1)|b|n}} || [[:Category:Williams prime PP|here]] ||[[Williams prime PP table|here]] || [[Williams prime PP least|here]] <ref>Values for bases ''b'' | + | | PP: {{Kbn|+|(b+1)|b|n}} || [[:Category:Williams prime PP|here]] ||[[Williams prime PP table|here]] || [[Williams prime PP least|here]] <ref>Values for bases ''b'' ≡ 1 mod 3 are always divisible by 3, so not listed here.</ref> |
|} | |} | ||
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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 Williams primes]: only Type MM {{Kbn|(b-1)|b|n}} for 3 ≤ ''b'' ≤ 2049, 1 ≤ ''n'' ≤ | + | *Steven Harvey: [http://harvey563.tripod.com/wills.txt Search for Williams primes]: only Type MM {{Kbn|(b-1)|b|n}} for 3 ≤ ''b'' ≤ 1019, 1 ≤ ''n'' ≤ 512 and 1020 ≤ ''b'' ≤ 2049, 1 ≤ ''n'' ≤ 100 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 ≤ ''n'' ≤ 1000 (mostly) and 1 ≤ ''b'' ≤ 1000 (2016) | *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) | *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) | ||
Revision as of 11:40, 4 June 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 | Table [1] | Smallest [2] |
|---|---|---|---|
| MM: (b-1)•bn-1 | here | here | here[3] |
| MP: (b-1)•bn+1 | here | here | here |
| PM: (b+1)•bn-1 | here | here | here |
| PP: (b+1)•bn+1 | here | here | here [4] |
Notes
- ↑ The table 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. For unknown values only the base is given.
- ↑ The list contains values for 2 ≤ b ≤ 2049.
- ↑ 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 3 ≤ b ≤ 1019, 1 ≤ n ≤ 512 and 1020 ≤ b ≤ 2049, 1 ≤ n ≤ 100 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 |
|
| Prime numbers |
|
