#Math #NT

# Fermet’s Little Theorem

If $p$ is a prime integer:

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

# Proof

Let $p$ be a prime integer. Say a necklace has $p$ beads and $a$ possible colors per bread. Except for a necklace with only one color, each combination of necklace colors has $p$ permutations. Therefore:

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