Lifting The Exponent Lemma for p=2/Corollary

Corollary to Lifting The Exponent Lemma for p=2

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

Let $n \geq1$ be an even natural number.

Then

$\nu_2 \left({x^n - y^n}\right) = \nu_2 \left({x + y}\right) + \nu_2 \left({x - y}\right) + \nu_2 \left({n}\right) - 1$

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

Proof

By Square Modulo 4, $4\mid x^2-y^2$.

By Lifting The Exponent Lemma for p=2:

\(\ds \nu_2 \left((x^2)^{n/2} - (y^2)^{n/2}\right)\)

\(=\)

\(\ds \nu_2 \left({x^2 - y^2}\right) + \nu_2 \left({n/2}\right)\)

\(\ds \)

\(=\)

\(\ds \nu_2 \left({x + y}\right) + \nu_2 \left({x - y}\right) + \nu_2 \left({n}\right) - 1\)

$\blacksquare$