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Difference between revisions of "Williams prime"
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! 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}} | ! 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}} || | + | | 2 || {{OEIS|A000043}} -> [[Williams prime MM 2|page]] || || {{OEIS|A002235}} -> [[Williams prime PM 2|page]] || {{OEIS|A002253}} -> [[Williams prime PP 2|page]] |
|- | |- | ||
| − | | 3 || {{OEIS|A003307}} || {{OEIS|A003306}} || {{OEIS|A005540}} || {{OEIS|A005537}} | + | | 3 || {{OEIS|A003307}} -> [[Williams prime MM 3|page]] || {{OEIS|A003306}} -> [[Williams prime MP 3|page]] || {{OEIS|A005540}} -> [[Williams prime PM 3|page]] || {{OEIS|A005537}} -> [[Williams prime PP 3|page]] |
|- | |- | ||
| − | | 4 || {{OEIS|A272057}} || | + | | 4 || {{OEIS|A272057}} -> [[Williams prime MM 4|page]] || {{OEIS|A326655}} -> [[Williams prime MP 4|page]] || || |
|- | |- | ||
| − | | 5 || {{OEIS|A046865}} || {{OEIS|A204322}} || {{OEIS|A257790}} || {{OEIS|A143279}} | + | | 5 || {{OEIS|A046865}} -> [[Williams prime MM 5|page]] || {{OEIS|A204322}} -> [[Williams prime MP 5|page]] || {{OEIS|A257790}} -> [[Williams prime PM 5|page]] || {{OEIS|A143279}} -> [[Williams prime PP 5|page]] |
|- | |- | ||
| − | | 6 || {{OEIS|A079906}} || {{OEIS|A247260}} || | + | | 6 || {{OEIS|A079906}} -> [[Williams prime MM 6|page]] || {{OEIS|A247260}} -> [[Williams prime MP 6|page]] || || |
|- | |- | ||
| − | | 7 || {{OEIS|A046866}} || {{OEIS|A245241}} || | + | | 7 || {{OEIS|A046866}} -> [[Williams prime MM 7|page]] || {{OEIS|A245241}} -> [[Williams prime MP 7|page]] || || |
|- | |- | ||
| − | | 8 || {{OEIS|A268061}} || {{OEIS|A269544}} || | + | | 8 || {{OEIS|A268061}} -> [[Williams prime MM 8|page]] || {{OEIS|A269544}} -> [[Williams prime MP 8|page]] || || |
|- | |- | ||
| − | | 9 || {{OEIS|A268356}} || {{OEIS|A056799}} || | + | | 9 || {{OEIS|A268356}} -> [[Williams prime MM 9|page]] || {{OEIS|A056799}} -> [[Williams prime MP 9|page]] || || |
|- | |- | ||
| − | | 10 || {{OEIS|A056725}} || {{OEIS|A056797}} || {{OEIS|A111391}} || | + | | 10 || {{OEIS|A056725}} -> [[Williams prime MM 10|page]] || {{OEIS|A056797}} -> [[Williams prime MP 10|page]] || {{OEIS|A111391}} -> [[Williams prime PM 10|page]] || |
| + | |- | ||
| + | | 11 || {{OEIS|A046867}} -> [[Williams prime MM 11|page]] || {{OEIS|A057462}} -> [[Williams prime MP 11|page]] || || | ||
| + | |- | ||
| + | | 12 || {{OEIS|A079907}} -> [[Williams prime MM 12|page]] || {{OEIS|A251259}} -> [[Williams prime MP 12|page]] || || | ||
| + | |- | ||
| + | | 13 || {{OEIS|A297348}} -> [[Williams prime MM 13|page]] || || || | ||
| + | |- | ||
| + | | 14 || {{OEIS|A273523}} -> [[Williams prime MM 14|page]] || || || | ||
|} | |} | ||
Revision as of 07:30, 5 August 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 -> page | A002235 -> page | A002253 -> page | |
| 3 | A003307 -> page | A003306 -> page | A005540 -> page | A005537 -> page |
| 4 | A272057 -> page | A326655 -> page | ||
| 5 | A046865 -> page | A204322 -> page | A257790 -> page | A143279 -> page |
| 6 | A079906 -> page | A247260 -> page | ||
| 7 | A046866 -> page | A245241 -> page | ||
| 8 | A268061 -> page | A269544 -> page | ||
| 9 | A268356 -> page | A056799 -> page | ||
| 10 | A056725 -> page | A056797 -> page | A111391 -> page | |
| 11 | A046867 -> page | A057462 -> page | ||
| 12 | A079907 -> page | A251259 -> page | ||
| 13 | A297348 -> page | |||
| 14 | A273523 -> page |
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 |
|
