proof of Wilson’s theorem result


set of primes We denote by the set of primes and by x¯ the multiplicative inverse of x in p.

Theorem (Generalisation of Wilson’s Theorem).

For all integers 1kp-1,pP(p-k)!(k-1)!(-1)k(modp)


If p is a prime, then:


and since (p-1)!-1(modp) (Wilson’s Theorem, simply pair up each number — except p-1 and 1, the only numbers in p which are their own inversesMathworldPlanetmathPlanetmathPlanetmathPlanetmath — with its inverse), the first implication follows.

Now, if p(p-1)!(k-1)!-(-1)k, then p as the opposite would mean that p=ab, for some integers 1<a,b<p, and so p would not be relatively prime to (p-1)!(k-1)! as the initial hypothesis implies. ∎

Title proof of Wilson’s theorem result
Canonical name ProofOfWilsonsTheoremResult
Date of creation 2013-03-22 15:07:08
Last modified on 2013-03-22 15:07:08
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 15
Author CWoo (3771)
Entry type Proof
Classification msc 11-00