
Glossary: Prime Pages: Top 5000: 
By Wilson’s theorem we know that every prime
p divides (p1)!+1. p is a
Wilson prime if p^{2} divides
(p1)!+1. For example 5 is a Wilson prime
because 25 divides 4!+1=25. The only known Wilson
primes are 5, 13, and 563; there are no others less
than 500,000,000.
It is conjectured that the number of Wilson primes is infinite and that the number of such primes between x and y should be about log(log y/log x). So it may be awhile before we find the fourth such prime! So what about composite numbers? To define a Wilson composite we first need an analog of Wilson's theorem that applies to composites: We say a composite number n is a Wilson Composite if n^{2} divides either m+1 or m1. The only such number below 50000 is 5971. Others include 558771, 1964215, 8121909 and 12326713; there are no others less than 10,000,000.
See Also: WilsonsTheorem, WieferichPrime, WallSunSunPrime Related pages (outside of this work) References:
Chris K. Caldwell © 19992016 (all rights reserved)
