Lifting The Exponent Lemma

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

Let $n \ge 1$ be a natural number.

Let $p$ be an odd prime.

Let:
 * $p \divides x - y$

and:
 * $p \nmid x y$

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

Then
 * $\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y} + \map {\nu_p} n$

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

Lemma
Let $k = \map {\nu_p} n$.

Then $n = p^k m$ such that $p \nmid m$.

By P-adic Valuation of Difference of Powers with Coprime Exponent:
 * $\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x^{p^k} - y^{p^k} }$

By repeatedly applying the lemma:
 * $\map {\nu_p} {x^{p^k} - y^{p^k} } = \map {\nu_p} {x - y} + k$

Also see

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