Wilson prime

Wilson Prime (named after English mathematician ) is a prime $p$ ${\ displaystyle p}$ $p$ such that $p^{2}$ ${\ displaystyle p ^ {2}}$ $p ^ {2}$ divides $(p-1)!+1$ ${\ displaystyle (p-1)! + 1}$ $(p-1)! + 1$ , where “!” means factorial . Note that by Wilson's theorem any prime $p$ ${\ displaystyle p}$ $p$ divides $(p-1)!+1$ ${\ displaystyle (p-1)! + 1}$ $(p-1)! + 1$ .

Only three Wilson primes are known - these are 5 , 13, and 563 (sequence A007540 in OEIS ). If others exist, they must be greater than 2⋅10 ¹³ . ^[one]

It has been hypothesized that there are infinitely many Wilson primes, and there are about log (log ( y ) / log ( x )) in the interval [ x , y ]. ^[2]

It has also been hypothesized (see comments on the sequence in OEIS) that p is the Wilson number if and only if:

\sum _{i=1}^{p-1}i^{p-1}=1^{p-1}+2^{p-1}+\cdots +(p-1)^{p-1}\equiv p-1{\pmod {p^{2}}}

{\ displaystyle \ sum _ {i = 1} ^ {p-1} i ^ {p-1} = 1 ^ {p-1} + 2 ^ {p-1} + \ cdots + (p-1) ^ {p-1} \ equiv p-1 {\ pmod {p ^ {2}}}}

{\ displaystyle \ sum _ {i = 1} ^ {p-1} i ^ {p-1} = 1 ^ {p-1} + 2 ^ {p-1} + \ cdots + (p-1) ^ {p-1} \ equiv p-1 {\ pmod {p ^ {2}}}}

.

Several attempts have been made to find Wilson primes. ^[3] ^[4] ^[5]

The Ibercivis Distributed Computing project includes a search for Wilson primes. ^[6] Another search is coordinated by the mersenneforum project. ^[7]

Content

Generalizations

Wilson's Almost Simple

Prime p for which (p - 1) holds! ≡ - 1 + Bp (mod p ² ) for small | B | can be called Wilson's almost simple . Almost Wilson primes with B = 0 are Wilson primes. The following table gives a list of all such numbers with | B | ≤ 100 from 10 ⁶ to 4⋅10 ¹¹ : ^[1]

p	B
1282279	+20
1306817	−30
1308491	−55
1433813	−32
1638347	−45
1640147	−88
1647931	+14
1666403	+99
1750901	+34
1851953	−50
2031053	−18
2278343	+21
2313083	+15
2695933	−73
3640753	+69
3677071	−32
3764437	−99
3958621	+75
5062469	+39
5063803	+40
6331519	+91
6706067	+45
7392257	+40
8315831	+3
8871167	−85
9278443	−75
9615329	+27
9756727	+23
10746881	−7
11465149	−62
11512541	−26
11892977	−7
12632117	−27
12893203	−53
14296621	+2
16711069	+95
16738091	+58
17879887	+63
19344553	−93
19365641	+75
20951477	+25
20972977	+58
21561013	−90
23818681	+23
27783521	−51
27812887	+21
29085907	+9
29327513	+13
30959321	+24
33187157	+60
33968041	+12
39198017	−7
45920923	−63
51802061	+4
53188379	−54
56151923	−1
57526411	−66
64197799	+13
72818227	−27
87467099	−2
91926437	−32
92191909	+94
93445061	−30
93559087	−3
94510219	−69
101710369	−70
111310567	+22
117385529	−43
176779259	+56
212911781	−92
216331463	−36
253512533	+25
282361201	+24
327357841	−62
411237857	−84
479163953	−50
757362197	−28
824846833	+60
866006431	−81
1227886151	−51
1527857939	−19
1636804231	+64
1686290297	+18
1767839071	+8
1913042311	−65
1987272877	+5
2100839597	−34
2312420701	−78
2476913683	+94
3542985241	−74
4036677373	−5
4271431471	+83
4296847931	+41
5087988391	+51
5127702389	+50
7973760941	+76
9965682053	−18
10242692519	−97
11355061259	−45
11774118061	−1
12896325149	+86
13286279999	+52
20042556601	+27
21950810731	+93
23607097193	+97
24664241321	+46
28737804211	−58
35525054743	+26
41659815553	+55
42647052491	+10
44034466379	+39
60373446719	−48
64643245189	−21
66966581777	+91
67133912011	+9
80248324571	+46
80908082573	−20
100660783343	+87
112825721339	+70
231939720421	+41
258818504023	+4
260584487287	−52
265784418461	−78
298114694431	+82

Wilson numbers

The Wilson number is an integer m such that W ( m ) ≡ 0 (mod m ), where W ( m ) means the Wilson fraction

W(m)={\frac {(m-1)!+1}{m}}

{\ displaystyle W (m) = {\ frac {(m-1)! + 1} {m}}}

(sequence A157250 in OEIS ).

If m is prime, then it will be Wilson's prime. Taking into account the number $one$ ${\ displaystyle 1}$ there are 13 Wilson numbers up to 5⋅10 ⁸ . ^[eight]

Notes

↑ ¹ ² A Search for Wilson primes Retrieved on November 2, 2012.
↑ The Prime Glossary: Wilson prime
↑ McIntosh, R. WILSON STATUS (Feb. 1999) (neopr.) . E-Mail to Paul Zimmermann (March 9, 2004). Date of treatment June 6, 2011. Archived January 29, 2013.
↑ A search for Wieferich and Wilson primes , p 443
↑ Ribenboim, P. Die Welt der Primzahlen: Geheimnisse und Rekorde : [] . - Berlin Heidelberg New York: Springer, 2006 .-- P. 241. - ISBN 3-540-34283-4 .
↑ Ibercivis site
↑ Distributed search for Wilson primes (at mersenneforum.org)
↑ Takashi Agoh; Karl Dilcher, Ladislav Skula. Wilson quotients for composite moduli (English) // Math. Comput. : journal. - 1998. - Vol. 67 , no. 222 . - P. 843-861 . - DOI : 10.1090 / S0025-5718-98-00951-X .

Links

NGWH Beeger. Quelques remarques sur les congruences r ^{p −1} ≡ 1 (mod p ² ) et ( p - 1!) ≡ −1 (mod p ² ) (Eng.) // Messenger of Mathematics : journal. - 1913-1914. - Vol. 43 . - P. 72-84 .
Karl Goldberg. A table of Wilson quotients and the third Wilson prime ( London ) // London Mathematical Society : journal. - 1953. - Vol. 28 , no. 2 . - P. 252-256 . - DOI : 10.1112 / jlms / s1-28.2.252 .
Paulo Ribenboim. The new book of prime number records. - Springer-Verlag , 1996. - P. 346. - ISBN 0-387-94457-5 .
Richard E. Crandall; Karl Dilcher, Carl Pomerance. A search for Wieferich and Wilson primes // Math. Comput. : journal. - 1997. - Vol. 66 , no. 217 . - P. 433-449 . - DOI : 10.1090 / S0025-5718-97-00791-6 .
Richard E. Crandall. Prime Numbers: A Computational Perspective. - Springer-Verlag, 2001. - P. 29. - ISBN 0-387-94777-9 .
Erna H. Pearson. On the Congruences ( p - 1)! ≡ −1 and 2 ^{p −1} ≡ 1 (mod p ² ) (English) // Math. Comput. : journal. - 1963. - Vol. 17 . - P. 194-195 .

Links

The Prime Glossary: Wilson prime
Weisstein, Eric W. Wilson prime on Wolfram MathWorld .
Status of the search for Wilson primes
Wilson Quotients for composite moduli
On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson

[Search-1] ¹ ² A Search for Wilson primes Retrieved on November 2, 2012.

[2] The Prime Glossary: Wilson prime

[3] McIntosh, R. WILSON STATUS (Feb. 1999) (neopr.) . E-Mail to Paul Zimmermann (March 9, 2004). Date of treatment June 6, 2011. Archived January 29, 2013.

[4] A search for Wieferich and Wilson primes , p 443

[5] Ribenboim, P. Die Welt der Primzahlen: Geheimnisse und Rekorde : [] . - Berlin Heidelberg New York: Springer, 2006 .-- P. 241. - ISBN 3-540-34283-4 .

[6] Ibercivis site

[7] Distributed search for Wilson primes (at mersenneforum.org)

[8] Takashi Agoh; Karl Dilcher, Ladislav Skula. Wilson quotients for composite moduli (English) // Math. Comput. : journal. - 1998. - Vol. 67 , no. 222 . - P. 843-861 . - DOI : 10.1090 / S0025-5718-98-00951-X .

Wilson prime

Content

Generalizations

Wilson's Almost Simple

Wilson numbers

See also

Notes

Links

Links

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