P-adic Valuation of Difference of Powers with Coprime Exponent/Proof 1
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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
- $\map {\nu_p} {x^n - y^n} = \map {\nu_p} {x - y}$
Proof
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 $p \nmid n$:
- $p \nmid n x^{n - 1}$
$\blacksquare$