Identity Relating Wilson's Theorem and Legendre Symbol
\[p\]
is an odd prime and \[gcd(a,p)=1\]
then \[(p-1)! \equiv -(a/p)a^{p-1} \; (mod \; p)\]
.By a property of the Legendre symbol,
\[(a/p) \equiv a^{\frac{p-1}{2}} \; (mod \; p)\]
. Hence \[(a/p)a^{\frac{p-1}{2}} \equiv (a/p)(a/p) \equiv 1 \; (mod \; p)\]
. From Wilson's Theorem \[(p-1)! \equiv -1 \; (mod \; p)\]
so \[(p-1)! \equiv -(a/p)a^{p-1} \; (mod \; p)\]
. 
