P-adic Valuation of Difference of Powers with Coprime Exponent/Proof 2
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Lemma
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 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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