P-adic Valuation of Difference of Powers with Coprime Exponent

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Theorem

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

Let $n \ge 1$ be a natural number.

Let $p$ be a prime number.

Let:

$p \divides x - y$

and:

$p \nmid x y n$.


Then

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

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


Proof 1

We have:

$x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$

We have to show that:

$p \nmid x^{n - 1} + \cdots + y^{n - 1}$

Because $x \equiv y \pmod p$:

$x^{n - 1} + \cdots + y^{n - 1} \equiv x^{n - 1} + x^{n - 1} + \cdots + x^{n - 1} = n x^{n - 1} \pmod p$

Because $p \nmid x$ and $p \nmid n$:

$p \nmid n x^{n - 1}$

$\blacksquare$


Proof 2

We have:

$x^n - y^n = \paren {x - y} \paren {x^{n - 1} + \cdots + y^{n - 1} }$

We have to prove that:

$p \nmid x^{n - 1} + \cdots + y^{n - 1}$

Let $\map P u = u^n - y^n$.

If $p \divides x^{n - 1} + \cdots + y^{n - 1}$, then $x$ would be a double root of $P$ modulo $p$.

By Double Root of Polynomial is Root of Derivative (or a version of this in modular arithmetic):

$p \divides \map {P'} x = n x^{n - 1}$

which is impossible.

Therefore:

$p \nmid x^{n - 1} + \cdots + y^{n - 1}$


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Sources

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