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

1. Wilson's Theorem - Brilliant

2. Euler's Theorem - Brilliant

3. Fermat's Little Theorem - Brilliant