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Fermet Euler Theorem.md

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+#Math #NT
+
+# Theorem
+
+Let $a$ and $m$ be coprime numbers.
+
+$$
+a^{\phi(m)} \equiv 1 \mod m
+$$
+
+This is a generalization of Fermet's Little Theorem, as $m$ is a prime number in Fermet’s Little Theorem.
+
+# Proof
+
+Let:
+
+$$
+A = \{p_1, p_2, p_3,... p_{\phi(m)} \} \mod m \\
+B = \{ap_1, ap_2, ap_3,...ap_{\phi(m)}\} \mod m
+$$
+
+Where $p_x$ is the $x$th number relatively prime to $m$.
+
+Since $a$ and $p_x$ are coprime to $m$, $ap_x$ is coprime to $m$. Since each $p_x$ is unique, $ap_x$ is unique, which makes set $B$ the same as set $A$.
+
+Since all terms are coprime to $m$:
+
+$$
+a^{\phi(m)} \prod _{k = 0}^{\phi(m)} p_k \equiv \prod _{k = 0}^{\phi(m)} p_k \mod m \\
+a^{\phi(m)} \equiv 1 \mod m
+$$