Lifting The Exponent Lemma for p=2

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

Let $n \geq1$ be a 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
Let $k=\nu_2(n)$.

Then $n=2^km$ with $2\nmid m$.

By Lifting The Exponent Lemma/Lemma 1:
 * $\nu_2(x^n-y^n) = \nu_2(x^{2^k} - y^{2^k})$

Note that $x^{2^k} - y^{2^k} = (x-y)\cdot \displaystyle\prod_{i=0}^{k-1}\left( x^{2^i}+y^{2^i} \right)$

By Square Modulo 4, $\nu_2(x^{2^i}+y^{2^i}) = 1$ for $i>0$.

Because $4\mid x-y$, $4\nmid x+y$.

Thus $\nu_2(x+y)=1$.

Thus

Also see

 * Lifting The Exponent Lemma
 * Lifting The Exponent Lemma for Sums for p=2