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,

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


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  .


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