Famous Theorems of Mathematics/Number Theory/Fermat's Little Theorem

Statement

If p is a rational prime, for all integers a ≠ 0,

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

Proofs

There are many proofs of Fermat's Little Theorem.

Proof 1 (Bijection)

Define a function $f(x)=ax$ (mod p). Let S={1,2,...,p-1} and T=f(S)={a,2a,...,(p-1)a}. We claim that these two sets are identical mod p.

Since all integers not equal to 0 have inverses mod p, for any integer m with 1≤m<p, $f(a^{-1}m)=m$. Then $f$ is surjective.

In addition, if $f(x)=f(y)$, then $ax\equiv ay$ and $a^{-1}ax\equiv x\equiv y\equiv a^{-1}ay$. Then $f$ is injective, and is bijective between S and T.

Then, mod p, the product of all of the elements of S will be equal to the product of elements of T, meaning that

$\prod _{k=1}^{p-1}k\equiv \prod _{k=1}^{p-1}ak{\pmod {p}}$ and that

$\prod _{k=1}^{p-1}k\equiv a^{p-1}\prod _{k=1}^{p-1}k{\pmod {p}}$.

Then

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