Difference between revisions of "Williams prime"

Revision as of 09:06, 14 June 2019

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.

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)`•`b`ⁿ-1	here	here	here^[5] 40 unknown	here
MP: `(b-1)`•`b`ⁿ+1	here	here	here 18 unknown
PM: `(b+1)`•`b`ⁿ-1	here	here	here
PP: `(b+1)`•`b`ⁿ+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.

Here are listed the available sequences in the On-Line Encyclopedia of Integer Sequences.

b	MM `(b-1)`•`b`ⁿ-1	MP `(b-1)`•`b`ⁿ+1	PM `(b+1)`•`b`ⁿ-1	PP `(b+1)`•`b`ⁿ+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

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

@@ Line 11: / Line 11: @@
 ! 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>
 |-
-| 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> || [[Williams prime MM remaining|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><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]] ||
+| 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 ||
 |-
 | PM: {{Kbn|(b+1)|b|n}} || [[:Category:Williams prime PM|here]] ||[[Williams prime PM table|here]] || [[Williams prime PM least|here]] ||