Lifting The Exponent Lemma

Theorem
Let $x, y \in \Z$ be integers.

Let $n \in \Z_{\ge 0}$ be a positive integer.

Let $p$ be an odd prime.

Let:
 * $p \mathrel \backslash x - y$

and:
 * $p \nmid x y$

where $\backslash$ and $\nmid$ denote divisibility and non-divisibility respectively.

Then
 * $\nu_p \left({x^n - y^n}\right) = \nu_p \left({x - y}\right) + \nu_p \left({n}\right)$

where $\nu_p$ denotes $p$-adic valuation.

Lemma 2
Let $k = \nu_p \left({n}\right)$.

Then $n = p^k m$ with $p \nmid m$.

By Lemma 1:
 * $\nu_p \left({x^n - y^n}\right) = \nu_p \left({x^{p^k} - y^{p^k} }\right)$

By repeatedly applying Lemma 2:
 * $\nu_p \left({x^{p^k} - y^{p^k} }\right) = \nu_p \left({x - y}\right) + k$

Also see

 * Lifting The Exponent Lemma for Sums
 * Lifting The Exponent Lemma for p=2