Lifting The Exponent Lemma for Sums for p=2

Theorem
Let $x, y \in \Z$ be integers with $x+y\neq0$.

Let $n \geq1$ be an odd 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.

Proof
This follows from the Lifting The Exponent Lemma for p=2 with $y$ replaced by $-y$.

Also see

 * Lifting The Exponent Lemma for Sums