Theorems of Wilson, Euler, and Fermat

Wilson's Theorem

A positive integer $$n > 1$$is a prime if and only if:

$$ (n-1)! \equiv -1 \mod n $$

Euler's Theorem

Let $$n \in \mathbb{Z}^{+}$$ and $$a \in \mathbb{Z}$$ s.t. $$gcd(a, n) = 1$$, then:

$$ a^{\phi(n)} \equiv 1 \mod n $$

Fermat's Little Theorem

Let $$p$$be a prime and $$a \in \mathbb{Z}$$, then:

$$ a^p \equiv a \mod p $$

or equivalently:

$$ a^{p-1} \equiv 1 \mod p $$

Reference