In number theory, a Wilson prime is a prime number divides " denotes the factorial function; compare this with Wilson's theorem. Both are named for 18th-century English mathematician John Wilson; in 1770, Edward Waring credited the theorem to Wilson, although it had been stated centuries earlier by Ibn al-Haytham.

The only known Wilson primes are 5, 13, and 563 is trivial", and credit the observation that 13 is a Wilson prime to . Early work on these numbers included searches by N. G. W. H. Beeger and Emma Lehmer, but 563 was not discovered until the early 1950s, when computer searches could be applied to the problem. If any others exist, they must be greater than 2 × 1013. It has been conjectured.

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

Generalizations

Wilson primes of order n

for every integer. Generalized Wilson primes of order are the primes .

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

are:

Near-Wilson primes

satisfying the congruence can be called a near-Wilson prime from up to 4.

Wilson numbers

A Wilson number term is positive if and only if has a primitive root and negative otherwise., and the quotients (called generalized Wilson quotients) are listed in . The Wilson numbers are

is a Wilson prime. There are 13 Wilson numbers up to 5.

See also

• PrimeGrid

• Table of congruences

• Wall–Sun–Sun prime

• Wieferich prime

• Wolstenholme prime

References

Further reading

External links

• The Prime Glossary: Wilson prime

• Status of the search for Wilson primes