Lifting The Exponent Lemma for Sums

Theorem

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

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

Let $p$ be an odd prime.

Let:

$p \divides x + y$

and:

$p \nmid x y$

where $\divides$ and $\nmid$ denote divisibility and non-divisibility respectively.

Then:

$\map {\nu_p} {x^n + y^n} = \map {\nu_p} {x + y} + \map {\nu_p} n$

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

Proof

This follows from the Lifting The Exponent Lemma with $y$ replaced by $-y$.

$\blacksquare$

Also see

• Lifting The Exponent Lemma for Sums for p=2