P-adic Valuation of Difference of Powers with Coprime Exponent

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

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

Let $p$ be a prime number.

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

and:
 * $p \nmid x y n$.

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

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

Proof
We have $x^n-y^n=(x-y)(x^{n-1}+\cdots+y^{n-1})$.

We have to prove that $p\nmid x^{n-1}+\cdots+y^{n-1}$.

Let $P(u)=u^n-y^n$.

If $p\mid x^{n-1}+\cdots+y^{n-1}$, $x$ would be a double root of $P$ modulo $p$.

By Double Root of Polynomial is Root of Derivative (or a version of this in modular arithmetic), $p\mid P'(x)=nx^{n-1}$, which is impossible.

Therefore, $p\nmid x^{n-1}+\cdots+y^{n-1}$.