User:Kip/Sandbox

Theorem
Let $x\in\Z_{>1}$ be a positive integer greater than one.

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

Then:
 * $\displaystyle A^{x}\equiv a \pmod m$

Is an $n^{th}$ root of unity modulo $m$ where:
 * $\displaystyle n=\frac{\phi(m)}{gcd(\phi(m),x)}$

$\phi(m)$ is Euler's Totient function of the modulus and $gcd(\phi(m),x)$ is the greatest common divisor of the totient and the power.

Proof
Let $\alpha=\frac{x}{gcd(\phi(m),x)}\in\Z_{>0}$ be a positive integer.
 * $\displaystyle A^{\alpha \phi(m)}\equiv a^n\equiv 1\,(mod\,m)$