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Difference between revisions of "Williams prime"
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| 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> || | | 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> || | ||
|} | |} | ||
| + | <references /> | ||
| − | == | + | ==Available Online Sequences== |
| − | < | + | Here are listed the available sequences in the [[On-Line Encyclopedia of Integer Sequences]]. |
| + | {| class="wikitable plainlinks" | ||
| + | ! b !! MM<br>{{Kbn|(b-1)|b|n}} !! MP<br>{{Kbn|+|(b-1)|b|n}} !! PM<br>{{Kbn|(b+1)|b|n}} !! PP<br>{{Kbn|+|(b+1)|b|n}} | ||
| + | |- | ||
| + | | 2 || {{OEIS|A000043}} || none || {{OEIS|A002235}} || {{OEIS|A002253}} | ||
| + | |- | ||
| + | | 3 || {{OEIS|A003307}} || {{OEIS|A003306}} || {{OEIS|A005540}} || {{OEIS|A005537}} | ||
| + | |- | ||
| + | | 4 || {{OEIS|A272057}} || none || none|| no primes | ||
| + | |- | ||
| + | | 5 || {{OEIS|A046865}} || {{OEIS|A204322}} || {{OEIS|A257790}} || {{OEIS|A143279}} | ||
| + | |- | ||
| + | | 6 || {{OEIS|A079906}} || {{OEIS|A247260}} || none || none | ||
| + | |- | ||
| + | | 7 || {{OEIS|A046866}} || {{OEIS|A245241}} || none || no primes | ||
| + | |- | ||
| + | | 8 || {{OEIS|A268061}} || {{OEIS|A269544}} || none || none | ||
| + | |- | ||
| + | | 9 || {{OEIS|A268356}} || {{OEIS|A056799}} || none || none | ||
| + | |- | ||
| + | | 10 || {{OEIS|A056725}} || {{OEIS|A056797}} || {{OEIS|A111391}} || no primes | ||
| + | |} | ||
==External links== | ==External links== | ||
Revision as of 09:33, 6 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[1] | Table [2] | Smallest [3] | Remaining[4] |
|---|---|---|---|---|
| MM: (b-1)•bn-1 | here | here | here[5] | 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 [6] |
- ↑ Containing all related pages for the type.
- ↑ 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.
- ↑ All data not yet available as separate page.
- ↑ The list contains values for 2 ≤ b ≤ 2049.
- ↑ Values for bases b ≡ 1 mod 3 are always divisible by 3, so not listed here.
Available Online Sequences
Here are listed the available sequences in the On-Line Encyclopedia of Integer Sequences.
| b | MM (b-1)•bn-1 |
MP (b-1)•bn+1 |
PM (b+1)•bn-1 |
PP (b+1)•bn+1 |
|---|---|---|---|---|
| 2 | A000043 | none | A002235 | A002253 |
| 3 | A003307 | A003306 | A005540 | A005537 |
| 4 | A272057 | none | none | no primes |
| 5 | A046865 | A204322 | A257790 | A143279 |
| 6 | A079906 | A247260 | none | none |
| 7 | A046866 | A245241 | none | no primes |
| 8 | A268061 | A269544 | none | none |
| 9 | A268356 | A056799 | none | none |
| 10 | A056725 | A056797 | A111391 | no primes |
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 ≤ 1024, 1 ≤ n ≤ 512 and 1025 ≤ 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 |
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| Prime numbers |
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