User:Kip/Sandbox

Definition
Let $n,m\in\Z_{>0}$ be positive integers

Then $a$ is said to be an $n^{th}$ root of unity modulo $m$ such that:
 * $a^n\equiv 1\,(mod\,m)$

Theorem
Let $a\in\Z$ be an $n^{th}$ root of unity modulo $m$

Let $A\in\Z_{>0}$ be a positive integer coprime with $m$

Let $\alpha\in\Z_{>0}$ be a positive integer

Let $n\in\Z_{>0}$ be a positive integer that is a factor of $\alpha\phi(m)$
 * $A^{\frac{\alpha\phi(m)}{n}}\equiv a\,(mod\,m)$

Proof

 * $A^{\alpha\phi(m)}\equiv a^n\,(mod\,m)$