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 $$a^{\phi(m)}(m)\equiv 1(m).$$

Proofs: 1

Proofs: 1

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  1. Landau, Edmund: "Vorlesungen ├╝ber Zahlentheorie, Aus der Elementaren Zahlentheorie", S. Hirzel, Leipzig, 1927