The following theorem was first proven by Pierre de Fermat (1601 - 1665). He proved it for prime numbers $p$ $$a^{p-1}(p)\equiv 1(p).$$ It is called Fermat’s little theorem to distinguish it from Fermat’s last theorem.

Later, this result was generalized by Euler, therefore, it is now called the Euler-Fermat theorem.

Theorem: Euler-Fermat Theorem 

Let $m > 1$ be a positive integer and let $\phi(m)$ denote the Euler function. For any integer $a\in\mathbb Z$ which is co-prime to $m$ we have the congruence $$a^{\phi(m)}(m)\equiv 1(m).$$

| | | | | created: 2019-05-11 18:13:05 | modified: 2019-06-22 08:44:06 | by: bookofproofs | references: [1272]

1.Proof: (related to "Euler-Fermat Theorem")