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==Definition==
 
==Definition==
−
A '''Williams number''' is a [[natural number]] of the form {{Kbn|(b-1)|b|n}} for integers ''b ≥ 2'' and ''n ≥ 1.
+
A '''Williams number''' is a [[natural number]] of the form {{Kbn|(b-1)|b|n}} for integers ''b'' ≥ 2 and ''n'' ≥ 1.
  
 
A '''Williams prime''' is a Williams number which is [[prime]].
 
A '''Williams prime''' is a Williams number which is [[prime]].
  
 
==Generalization==
 
==Generalization==
−
Varying both signs, there're four different types of numbers similiar to Williams numbers:
+
Varying both signs, there're four different types of numbers similiar to Williams numbers.
−
*[[:Category:Williams prime MM|Type {{Kbn|(b-1)|b|n}}]]
+
 
−
*[[:Category:Williams prime MP|Type {{Kbn|+|(b-1)|b|n}}]]
+
Lists of primes for bases ''b'' and ''n''-values can be found here:
−
*[[:Category:Williams prime PM|Type {{Kbn|(b+1)|b|n}}]]
+
{| class="wikitable"
−
*[[:Category:Williams prime PP|Type {{Kbn|+|(b+1)|b|n}}]]
+
! 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]]<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]]<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]]<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]]<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 />
 +
 
 +
==Available Online Sequences==
 +
Here are listed the available sequences in the [[On-Line Encyclopedia of Integer Sequences]].
 +
{| class="wikitable plainlinks"
 +
! 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}} -> [[Williams prime MM 2|page]] || || {{OEIS|A002235}} -> [[Williams prime PM 2|page]] || {{OEIS|A002253}} -> [[Williams prime PP 2|page]]
 +
|-
 +
| 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}} -> [[Williams prime MM 4|page]] || {{OEIS|A326655}} -> [[Williams prime MP 4|page]] || ||
 +
|-
 +
| 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}} -> [[Williams prime MM 6|page]] || {{OEIS|A247260}} -> [[Williams prime MP 6|page]] || ||
 +
|-
 +
| 7 || {{OEIS|A046866}} -> [[Williams prime MM 7|page]] || {{OEIS|A245241}} -> [[Williams prime MP 7|page]] || ||
 +
|-
 +
| 8 || {{OEIS|A268061}} -> [[Williams prime MM 8|page]] || {{OEIS|A269544}} -> [[Williams prime MP 8|page]] || ||
 +
|-
 +
| 9 || {{OEIS|A268356}} -> [[Williams prime MM 9|page]] || {{OEIS|A056799}} -> [[Williams prime MP 9|page]] || ||
 +
|-
 +
| 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]] || || ||
 +
|}
  
 
==External links==
 
==External links==
 
*H. C. Williams: [http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf "The primality of certain integers of the form 2Ar^n-1"], Acta Arith. 39 (1981), 7-17
 
*H. C. Williams: [http://matwbn.icm.edu.pl/ksiazki/aa/aa39/aa3912.pdf "The primality of certain integers of the form 2Ar^n-1"], Acta Arith. 39 (1981), 7-17
 
*A. Stein, H. C. Williams: [https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf "Explicit primality criteria for (p−1)p<sup>n</sup>−1"], Math. Comp. 69 (2000), 1721-1734
 
*A. Stein, H. C. Williams: [https://www.ams.org/journals/mcom/2000-69-232/S0025-5718-00-01212-6/S0025-5718-00-01212-6.pdf "Explicit primality criteria for (p−1)p<sup>n</sup>−1"], Math. Comp. 69 (2000), 1721-1734
−
*Steven Harvey: [http://harvey563.tripod.com/wills.txt Search for original Williams primes]
+
*Steven Harvey: [http://harvey563.tripod.com/wills.txt Search for Williams primes]: only Type MM {{Kbn|(b-1)|b|n}} for 3 ≤ ''b'' ≤ 1024, 1 ≤ ''n'' ≤ 512 and 1025 ≤ ''b'' ≤ 2049, 1 ≤ ''n'' ≤ 100 and some higher (2006-2019)
 +
*Mauro Fiorentini: [http://www.bitman.name/math/table/484 Type MM], [http://www.bitman.name/math/table/477 Type MP], [http://www.bitman.name/math/table/471 Type PM], [http://www.bitman.name/math/table/474 Type PP] for 0 ≤ ''n'' ≤ 1000 (mostly) and 1 ≤ ''b'' ≤ 1000 (2016)
 +
*Eric Chen: [https://www.mersenneforum.org/showthread.php?t=21818 Thread] at [[MersenneForum]] including dual forms for 0 ≤ ''n'' ≤ 5000 and 1 ≤ ''b'' ≤ 64 and some higher (2016-2019)
 
*[[Wikipedia:Williams number|Williams number]]
 
*[[Wikipedia:Williams number|Williams number]]
 +
{{Navbox Williams primes}}
 
{{Navbox NumberClasses}}
 
{{Navbox NumberClasses}}
 
[[Category:Williams prime| ]]
 
[[Category:Williams prime| ]]

Latest revision as of 10:14, 26 August 2024

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
  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.

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

Williams primes
Number classes
General numbers
Special numbers
Prime numbers
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