Named after John Wilson 3 5, 13, 563 A007540Wilson primes: primes ${\displaystyle p}$ such that ${\displaystyle (p-1)!\equiv -1\ (\operatorname {mod} {p^{2)))}$

In number theory, a Wilson prime is a prime number ${\displaystyle p}$ such that ${\displaystyle p^{2))$ divides ${\displaystyle (p-1)!+1}$, where "${\displaystyle !}$" denotes the factorial function; compare this with Wilson's theorem, which states that every prime ${\displaystyle p}$ divides ${\displaystyle (p-1)!+1}$. 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.[2]

The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). Costa et al. write that "the case ${\displaystyle p=5}$ is trivial", and credit the observation that 13 is a Wilson prime to Mathews (1892).[3][4] Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer,[5][3][6] but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem.[3][7][8] If any others exist, they must be greater than 2 × 1013.[3] It has been conjectured that infinitely many Wilson primes exist, and that the number of Wilson primes in an interval ${\displaystyle [x,y]}$ is about ${\displaystyle \log \log _{x}y}$.[9]

Several computer searches have been done in the hope of finding new Wilson primes.[10][11][12] The Ibercivis distributed computing project includes a search for Wilson primes.[13] Another search was coordinated at the Great Internet Mersenne Prime Search forum.[14]

## Generalizations

### Wilson primes of order n

Wilson's theorem can be expressed in general as ${\displaystyle (n-1)!(p-n)!\equiv (-1)^{n}\ {\bmod {p))}$ for every integer ${\displaystyle n\geq 1}$ and prime ${\displaystyle p\geq n}$. Generalized Wilson primes of order n are the primes p such that ${\displaystyle p^{2))$ divides ${\displaystyle (n-1)!(p-n)!-(-1)^{n))$.

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

The smallest generalized Wilson primes of order ${\displaystyle n}$ are:

5, 2, 7, 10429, 5, 11, 17, ... (The next term > 1.4 × 107) (sequence A128666 in the OEIS)

### Near-Wilson primes

A prime ${\displaystyle p}$ satisfying the congruence ${\displaystyle (p-1)!\equiv -1+Bp\ (\operatorname {mod} {p^{2)))}$ with small ${\displaystyle |B|}$ can be called a near-Wilson prime. Near-Wilson primes with ${\displaystyle B=0}$ are bona fide Wilson primes. The table on the right lists all such primes with ${\displaystyle |B|\leq 100}$ from 106 up to 4×1011.[3]

### Wilson numbers

A Wilson number is a natural number ${\displaystyle n}$ such that ${\displaystyle W(n)\equiv 0\ (\operatorname {mod} {n^{2)))}$, where

${\displaystyle W(n)=\prod _{\stackrel {1\leq k\leq n}{\gcd(k,n)=1)){k}\pm 1,}$
and where the ${\displaystyle \pm 1}$ term is positive if and only if ${\displaystyle n}$ has a primitive root and negative otherwise.[15] For every natural number ${\displaystyle n}$, ${\displaystyle W(n)}$ is divisible by ${\displaystyle n}$, and the quotients (called generalized Wilson quotients) are listed in . The Wilson numbers are

1, 5, 13, 563, 5971, 558771, 1964215, 8121909, 12326713, 23025711, 26921605, 341569806, 399292158, ... (sequence A157250 in the OEIS)

If a Wilson number ${\displaystyle n}$ is prime, then ${\displaystyle n}$ is a Wilson prime. There are 13 Wilson numbers up to 5×108.[16]

## 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. ^ O'Connor, John J.; Robertson, Edmund F. "Abu Ali al-Hasan ibn al-Haytham". MacTutor History of Mathematics archive. University of St Andrews.
3. Costa, Edgar; Gerbicz, Robert; Harvey, David (2014). "A search for Wilson primes". Mathematics of Computation. 83 (290): 3071–3091. arXiv:1209.3436. doi:10.1090/S0025-5718-2014-02800-7. MR 3246824.
4. ^ Mathews, George Ballard (1892). "Example 15". Theory of Numbers, Part 1. Deighton & Bell. p. 318.
5. ^ Lehmer, Emma (April 1938). "On congruences involving Bernoulli numbers and the quotients of Fermat and Wilson" (PDF). Annals of Mathematics. 39 (2): 350–360. doi:10.2307/1968791. JSTOR 1968791. Retrieved 8 March 2011.
6. ^ Beeger, N. G. W. H. (1913–1914). "Quelques remarques sur les congruences ${\displaystyle r^{p-1}\equiv 1\ (\operatorname {mod} {p^{2)))}$ et ${\displaystyle (p-1)!\equiv -1\ (\operatorname {mod} {p^{2)))}$". The Messenger of Mathematics. 43: 72–84.
7. ^ Wall, D. D. (October 1952). "Unpublished mathematical tables" (PDF). Mathematical Tables and Other Aids to Computation. 6 (40): 238. doi:10.2307/2002270. JSTOR 2002270.
8. ^ Goldberg, Karl (1953). "A table of Wilson quotients and the third Wilson prime". J. London Math. Soc. 28 (2): 252–256. doi:10.1112/jlms/s1-28.2.252.
9. ^ The Prime Glossary: Wilson prime
10. ^ McIntosh, R. (9 March 2004). "WILSON STATUS (Feb. 1999)". E-Mail to Paul Zimmermann. Retrieved 6 June 2011.
11. ^ Crandall, Richard E.; Dilcher, Karl; Pomerance, Carl (1997). "A search for Wieferich and Wilson primes". Math. Comput. 66 (217): 433–449. Bibcode:1997MaCom..66..433C. doi:10.1090/S0025-5718-97-00791-6. See p. 443.
12. ^ Ribenboim, P.; Keller, W. (2006). Die Welt der Primzahlen: Geheimnisse und Rekorde (in German). Berlin Heidelberg New York: Springer. p. 241. ISBN 978-3-540-34283-0.
13. ^ "Ibercivis site". Archived from the original on 2012-06-20. Retrieved 2011-03-10.
14. ^ Distributed search for Wilson primes (at mersenneforum.org)
15. ^
16. ^ Agoh, Takashi; Dilcher, Karl; Skula, Ladislav (1998). "Wilson quotients for composite moduli" (PDF). Math. Comput. 67 (222): 843–861. Bibcode:1998MaCom..67..843A. doi:10.1090/S0025-5718-98-00951-X.