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

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


There are many proofs of Fermat's Little Theorem.

Proof 1 (Bijection)

Define a function (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, . Then is surjective.

In addition, if , then and . Then 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

and that