Lifting The Exponent Lemma for Sums for p=2

Theorem
Let $x, y \in \Z$ be integers with $x + y \ne 0$.

Let $n \ge 1$ be an odd natural number.

Let:
 * $2 \divides x + y$

where $\divides$ denotes divisibility.

Then:
 * $\map {\nu_2} {x^n + y^n} = \map {\nu_2} {x + y}$

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