public-notes/Fermat’s Little Theorem.md

#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