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Difference between revisions of "Williams prime"
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! Type !! Category !! List<br>table <ref>The list table of any type contains only bases which are included as a separate page.</ref> !! List<br>smallest <ref>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.</ref> | ! Type !! Category !! List<br>table <ref>The list table of any type contains only bases which are included as a separate page.</ref> !! List<br>smallest <ref>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.</ref> | ||
|- | |- | ||
| − | | MM: {{Kbn|(b-1)|b|n}} || [[:Category:Williams prime MM|here]] ||[[Williams prime MM table|here]] || [[ | + | | MM: {{Kbn|(b-1)|b|n}} || [[:Category:Williams prime MM|here]] ||[[Williams prime MM table|here]] || [[Williams prime MM least|here]] |
|- | |- | ||
| MP: {{Kbn|+|(b-1)|b|n}} || [[:Category:Williams prime MP|here]] ||[[Williams prime MP table|here]] || [[:File:Williams prime MP smallest.txt|here]] | | MP: {{Kbn|+|(b-1)|b|n}} || [[:Category:Williams prime MP|here]] ||[[Williams prime MP table|here]] || [[:File:Williams prime MP smallest.txt|here]] | ||
Revision as of 13:26, 15 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 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 |
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| Prime numbers |
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