Wilson prime explained

Named After:	John Wilson
Terms Number:	3
First Terms:	5, 13, 563
Oeis:	A007540
Oeis Name:	Wilson primes: primes p such that (p -1 )!\equiv -1 (\operatorname{mod}{p ^{2})}

p such that p ^{2} divides (p -1 )! +1 , where "! " denotes the factorial function; compare this with Wilson's theorem, which states that every prime p divides (p -1 )! +1 . Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson,^[1] although it had been stated centuries earlier by Ibn al-Haytham.

The only known Wilson primes are 5, 13, and 563 . Costa et al. write that "the case

p =5 is trivial", and credit the observation that 13 is a Wilson prime to .^[2] Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer,^{[3] [4]} but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.^{[5] [6]} If any others exist, they must be greater than 2 × 10¹³.^[7] It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval [x,y] is about loglog _{x} y .^[8]

Several computer searches have been done in the hope of finding new Wilson primes.^{[9] [10] [11]} The Ibercivis distributed computing project includes a search for Wilson primes.^[12] Another search was coordinated at the Great Internet Mersenne Prime Search forum.^[13]

Generalizations

Wilson primes of order

Wilson's theorem can be expressed in general as

(n -1 )!(p - n)!\equiv( -1 ) ^{n \bmod} p for every integer n \ge 1 and prime p \ge n . Generalized Wilson primes of order are the primes such that p ^{2} divides (n -1 )!(p - n)! - ( -1 ) ^{n} .

It was conjectured that for every natural number, there are infinitely many Wilson primes of order .

The smallest generalized Wilson primes of order

n are:

Near-Wilson primes

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

A prime p satisfying the congruence (p -1 )!\equiv -1+ Bp (\operatorname{mod}{p ^{2})} with small |B| can be called a near-Wilson prime. Near-Wilson primes with B =0 are bona fide Wilson primes. The table on the right lists all such primes with |B|\le 100 from up to 4.^[7]

Wilson numbers

A Wilson number is a natural number

n such that W(n)\equiv 0 (\operatorname{mod}{n ^{2})} , where $$W(n)\; =\; \backslash pm1+\backslash prod\_\backslash stackrel,$$and where the \pm 1 term is positive if and only if n has a primitive root and negative otherwise.^[14] For every natural number n , W(n) is divisible by n , and the quotients (called generalized Wilson quotients) are listed in . The Wilson numbers are

If a Wilson number

n is prime, then n is a Wilson prime. There are 13 Wilson numbers up to 5.^[15]

See also

• PrimeGrid
• Table of congruences
• Wall–Sun–Sun prime
• Wieferich prime
• Wolstenholme prime

Further reading

• Book: Prime Numbers: A Computational Perspective . Richard E. . Crandall . Carl . Pomerance . Springer-Verlag . 2001 . 29 . 978-0-387-94777-8 .
• Erna H. . Pearson . On the Congruences (p − 1)! ≡ −1 and 2^p−1 ≡ 1 (mod p²) . Math. Comput. . 17 . 194–195 . 1963 .

External links

• The Prime Glossary: Wilson prime
  • Status of the search for Wilson primes

Notes and References

1. Edward Waring, Meditationes Algebraicae (Cambridge, England: 1770), page 218 (in Latin). In the third (1782) edition of Waring's Meditationes Algebraicae, Wilson's theorem appears as problem 5 on page 380. On that page, Waring states: "Hanc maxime elegantem primorum numerorum proprietatem invenit vir clarissimus, rerumque mathematicarum peritissimus Joannes Wilson Armiger." (A man most illustrious and most skilled in mathematics, Squire John Wilson, found this most elegant property of prime numbers.)
2. Book: Mathews, George Ballard. Example 15. 318. Theory of Numbers, Part 1. George Ballard Mathews. 1892. Deighton & Bell.
3. 10.2307/1968791 . Lehmer . Emma . Emma Lehmer . On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson . . 39 . 2 . 350–360 . April 1938 . 8 March 2011. 1968791 .
4. N. G. W. H. . Beeger . N. G. W. H. Beeger . Quelques remarques sur les congruences r ^{p-1} \equiv 1 (\operatorname{mod}{p ^{2})} et (p -1 )!\equiv -1 (\operatorname{mod}{p ^{2})} . . 43 . 72–84 . 1913–1914.
5. Wall . D. D. . Donald Dines Wall . October 1952 . 10.2307/2002270 . 40 . Mathematical Tables and Other Aids to Computation . 2002270 . 238 . Unpublished mathematical tables . 6.
6. Karl . Goldberg . A table of Wilson quotients and the third Wilson prime . J. London Math. Soc.. 28 . 2 . 252–256 . 1953 . 10.1112/jlms/s1-28.2.252 .
7. Costa . Edgar . Gerbicz . Robert . Harvey . David . 1209.3436 . 10.1090/S0025-5718-2014-02800-7 . 290 . Mathematics of Computation . 3246824 . 3071–3091 . A search for Wilson primes . 83 . 2014. 6738476 .
8. http://primes.utm.edu/glossary/page.php?sort=WilsonPrime The Prime Glossary: Wilson prime
9. Web site: McIntosh . R. . Richard McIntosh . WILSON STATUS (Feb. 1999) . . 9 March 2004 . 6 June 2011.
10. Richard E. . Crandall . Karl . Dilcher . Carl . Pomerance . A search for Wieferich and Wilson primes . Math. Comput. . 66 . 217 . 433–449 . 1997 . 10.1090/S0025-5718-97-00791-6 . 1997MaCom..66..433C . free . See p. 443.
11. Book: Ribenboim . P. . Paulo Ribenboim . Keller . W. . Die Welt der Primzahlen: Geheimnisse und Rekorde . Springer . 2006 . Berlin Heidelberg New York . 241 . de . 978-3-540-34283-0.
12. Web site: Ibercivis site . 2011-03-10 . 2012-06-20 . https://web.archive.org/web/20120620210249/http://www.ibercivis.net/index.php?module=public&section=channels&action=view&id_channel=2&id_subchannel=138 . dead .
13. http://www.mersenneforum.org/showthread.php?t=16028 Distributed search for Wilson primes
14. see Gauss's generalization of Wilson's theorem
15. 10.1090/S0025-5718-98-00951-X . Takashi . Agoh . Karl . Dilcher . Ladislav . Skula . Ladislav Skula. Wilson quotients for composite moduli . Math. Comput. . 67 . 222 . 843–861 . 1998 . 1998MaCom..67..843A . free .