Lifting The Exponent Lemma

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Theorem

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

Let $n \geq1$ be a natural number.

Let $p$ be an odd prime.

Let:

$p \mathrel \backslash x - y$

and:

$p \nmid x y$

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


Then

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

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


Proof

Lemma

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

Let $p$ be an odd prime.

Let:

$p \mathrel \backslash x - y$

and:

$p \nmid x y$.


Then

$\nu_p \left({x^p - y^p}\right) = \nu_p \left({x - y}\right) + 1$



Let $k = \nu_p \left({n}\right)$.

Then $n = p^k m$ with $p \nmid m$.

By P-adic Valuation of Difference of Powers with Coprime Exponent:

$\nu_p \left({x^n - y^n}\right) = \nu_p \left({x^{p^k} - y^{p^k} }\right)$

By repeatedly applying the Lemma:

$\nu_p \left({x^{p^k} - y^{p^k} }\right) = \nu_p \left({x - y}\right) + k$

$\blacksquare$


Also see


Sources