Fermat’s Little Theorem

Related Concepts

Euler’s Totient Function; Fermat Primality Test; Modular Arithmetic; Number Theory; Prime Number

Definition

Fermat’s little theorem states that if p is a prime number and a is any number not divisible by p, then ${a}^{p-1} \equiv 1 \mathbin{\rm mod}\,\,p$.

Background

Pierre de Fermat (1601–1665) was one of the most renowned mathematicians in history. He focused much of his work on Number Theory, though he made great contributions to many other areas of mathematics. Fermat’s famous “last theorem” was a remark Fermat made in a margin of a book, for which he claimed to have a proof but the margin was too small to write it down, and remained an open problem for over three centuries.

Fermat’s little theorem was first stated in a letter to Frénicle de Bessy in 1640, though it was not called “Fermat’s little theorem” until the twentieth century.

Theory

The proof follows easily from the following observations. Consider the product $(a)(2a)(3a)\ldots ((p - 1)a)$, then on...