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 $-y$.

$\blacksquare$

Also see

• Lifting The Exponent Lemma for Sums