Wilson prime
Type of prime number / From Wikipedia, the free encyclopedia
In number theory, a Wilson prime is a prime number such that
divides
, where " !}
" denotes the factorial function; compare this with Wilson's theorem, which states that every prime
divides
. 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]
Named after | John Wilson |
---|---|
No. of known terms | 3 |
First terms | 5, 13, 563 |
OEIS index |
|
The only known Wilson primes are 5, 13, and 563 (sequence A007540 in the OEIS). Costa et al. write that "the case 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
is about
.[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]