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> || [[Williams prime MM remaining|here]]
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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]]
 
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| MP: {{Kbn|+|(b-1)|b|n}} || [[:Category:Williams prime MP|here]] ||[[Williams prime MP table|here]] || [[Williams prime MP least|here]] ||
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| 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 ||
 
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|-
 
| PM: {{Kbn|(b+1)|b|n}} || [[:Category:Williams prime PM|here]] ||[[Williams prime PM table|here]] || [[Williams prime PM least|here]] ||
 
| PM: {{Kbn|(b+1)|b|n}} || [[:Category:Williams prime PM|here]] ||[[Williams prime PM table|here]] || [[Williams prime PM least|here]] ||

Revision as of 09:06, 14 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]
40 unknown
here
MP: (b-1)•bn+1 here here here
21 unknown
PM: (b+1)•bn-1 here here here
PP: (b+1)•bn+1 here here here [6]
  1. Containing all related pages for the type.
  2. The table contains only bases which are included as a separate page.
  3. The list of smallest primes of any base is an ASCII file for 2 ≤ b ≤ 1024. For unknown values only the base is given.
  4. All data not yet available as separate page.
  5. The list contains values for 2 ≤ b ≤ 2049.
  6. 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.

External links

Number classes
General numbers
Special numbers
Prime numbers