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>  
    
−   MM: {{Kbn(b1)bn}}  [[:Category:Williams prime MMhere]] [[Williams prime MM tablehere]]  [[Williams prime MM leasthere]]<ref>The list contains values for 2 ≤ ''b'' ≤ 2049.</ref>  [[Williams prime MM remaininghere]]  +   MM: {{Kbn(b1)bn}}  [[:Category:Williams prime MMhere]] [[Williams prime MM tablehere]]  [[Williams prime MM leasthere]]<ref>The list contains values for 2 ≤ ''b'' ≤ 2049.</ref><br>{{PAGESINCATEGORY:Williams prime MM without}} unknown  [[Williams prime MM remaininghere]] 
    
−   MP: {{Kbn+(b1)bn}}  [[:Category:Williams prime MPhere]] [[Williams prime MP tablehere]]  [[Williams prime MP leasthere]]   +   MP: {{Kbn+(b1)bn}}  [[:Category:Williams prime MPhere]] [[Williams prime MP tablehere]]  [[Williams prime MP leasthere]]<br>{{PAGESINCATEGORY:Williams prime MP without}} unknown  
    
 PM: {{Kbn(b+1)bn}}  [[:Category:Williams prime PMhere]] [[Williams prime PM tablehere]]  [[Williams prime PM leasthere]]    PM: {{Kbn(b+1)bn}}  [[:Category:Williams prime PMhere]] [[Williams prime PM tablehere]]  [[Williams prime PM leasthere]]  
Revision as of 09:06, 14 June 2019
Definition
A Williams number is a natural number of the form (b1)•b^{n}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 nvalues can be found here:
Type  Category^{[1]}  Table ^{[2]}  Smallest ^{[3]}  Remaining^{[4]} 

MM: (b1)•b^{n}1  here  here  here^{[5]} 40 unknown 
here 
MP: (b1)•b^{n}+1  here  here  here 22 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.
Available Online Sequences
Here are listed the available sequences in the OnLine Encyclopedia of Integer Sequences.
b  MM (b1)•b^{n}1 
MP (b1)•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 
External links
 H. C. Williams: "The primality of certain integers of the form 2Ar^n1", Acta Arith. 39 (1981), 717
 A. Stein, H. C. Williams: "Explicit primality criteria for (p−1)p^{n}−1", Math. Comp. 69 (2000), 17211734
 Steven Harvey: Search for Williams primes: only Type MM (b1)•b^{n}1 for 3 ≤ b ≤ 1024, 1 ≤ n ≤ 512 and 1025 ≤ b ≤ 2049, 1 ≤ n ≤ 100 and some higher (20062019)
 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 (20162019)
 Williams number
Number classes
General numbers 
Special numbers 
Prime numbers 
