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Difference between revisions of "Williams prime"
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! Type !! Category<ref>Containing all related pages for the type.</ref> !! 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> !! Remaining<ref>All data not yet available as separate page.</ref> | ! Type !! Category<ref>Containing all related pages for the type.</ref> !! 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> !! Remaining<ref>All data not yet available as separate page.</ref> | ||
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| − | | 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><br>{{PAGESINCATEGORY:Williams prime MM without}} unknown || [[Williams prime MM remaining|here]] | + | | MM: {{Kbn|(b-1)|b|n}} || [[:Category:Williams prime MM|here]] ||[[Williams prime MM table|here]]<br>{{#expr:{{PAGESINCATEGORY:Williams prime MM|pages}}-3}} bases || [[Williams prime MM least|here]]<ref>The list contains values for 2 ≤ ''b'' ≤ 2049.</ref><br>{{PAGESINCATEGORY:Williams prime MM without}} unknown || [[Williams prime MM remaining|here]] |
|- | |- | ||
| − | | MP: {{Kbn|+|(b-1)|b|n}} || [[:Category:Williams prime MP|here]] ||[[Williams prime MP table|here]] || [[Williams prime MP least|here]]<br>{{PAGESINCATEGORY:Williams prime MP without}} unknown || | + | | MP: {{Kbn|+|(b-1)|b|n}} || [[:Category:Williams prime MP|here]] ||[[Williams prime MP table|here]]<br>{{#expr:{{PAGESINCATEGORY:Williams prime MP|pages}}-3}} bases || [[Williams prime MP least|here]]<br>{{PAGESINCATEGORY:Williams prime MP without}} unknown || [[Williams prime MP remaining|here]] |
|- | |- | ||
| − | | PM: {{Kbn|(b+1)|b|n}} || [[:Category:Williams prime PM|here]] ||[[Williams prime PM table|here]] || [[Williams prime PM least|here]]<br>{{PAGESINCATEGORY:Williams prime PM without}} unknown || | + | | PM: {{Kbn|(b+1)|b|n}} || [[:Category:Williams prime PM|here]] ||[[Williams prime PM table|here]]<br>{{#expr:{{PAGESINCATEGORY:Williams prime PM|pages}}-3}} bases || [[Williams prime PM least|here]]<br>{{PAGESINCATEGORY:Williams prime PM without}} unknown || [[Williams prime PM remaining|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'' ≡ 1 mod 3 are always divisible by 3, so not listed here.</ref><br>{{PAGESINCATEGORY:Williams prime PP without}} unknown || | + | | PP: {{Kbn|+|(b+1)|b|n}} || [[:Category:Williams prime PP|here]] ||[[Williams prime PP table|here]]<br>{{#expr:{{PAGESINCATEGORY:Williams prime PP|pages}}-3}} bases || [[Williams prime PP least|here]] <ref>Values for bases ''b'' ≡ 1 mod 3 are always divisible by 3, so not listed here.</ref><br>{{PAGESINCATEGORY:Williams prime PP without}} unknown || [[Williams prime PP remaining|here]] |
|} | |} | ||
<references /> | <references /> | ||
Revision as of 08:19, 18 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 214 bases |
here[5] 40 unknown |
here |
| MP: (b-1)•bn+1 | here | here 189 bases |
here 18 unknown |
here |
| PM: (b+1)•bn-1 | here | here 128 bases |
here 2 unknown |
here |
| PP: (b+1)•bn+1 | here | here 110 bases |
here [6] 4 unknown |
here |
- ↑ 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 |
|
| Prime numbers |
|
