Lifting The Exponent Lemma for p=2

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

Let $n \geq1$ be a natural number.

Let:
 * $4 \mathrel \backslash x - y$

where $\backslash$ denotes divisibility.

Then
 * $\nu_2 \left({x^n - y^n}\right) = \nu_2 \left({x - y}\right) + \nu_2 \left({n}\right)$

where $\nu_2$ denotes $2$-adic valuation.

Also see

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