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 $k=\nu_p(n)$.

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

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

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

Also see

 * Lifting The Exponent Lemma for Sums