Lifting The Exponent Lemma/Lemma
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Lemma
Let $x, y \in \Z$ be distinct integers.
Let $p$ be an odd prime.
Let:
- $p \divides x - y$
and:
- $p \nmid x y$.
Then
- $\map {\nu_p} {x^p - y^p} = \map {\nu_p} {x - y} + 1$
where $\nu_p$ denotes $p$-adic valuation.
Proof
Let $\map {\nu_p} {x - y} = k$.
Then $x=p^k m + y$ where $p \nmid m$.
We have:
\(\ds x^p - y^p\) | \(=\) | \(\ds (p^k m + y)^p - y^p\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 0}^p \paren {\binom p i \paren {p^k m}^{p - i} y^i} - y^p\) | Binomial Theorem | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 0}^{p - 2} \paren {\binom p i \paren {p^k m}^{p - i} y^i} + \binom p {p - 1} \paren {p^k m} y^{p - 1}\) | picking out the last two terms from the summation | |||||||||||
\(\ds \) | \(=\) | \(\ds \sum_{i \mathop = 0}^{p - 2} \paren {\binom p i \paren {p^k m}^{p - i} y^i} + p^{k + 1} m y^{p - 1}\) |
Note that all terms in the above expression have a factor of $p$ to the order at least $k+1$.
So, $p^{k + 1} \mid x^p - y^p$.
Also note that all terms in the summation have a factor of $p$ to the order at least $k + 2$.
But in the term $p^{k + 1} m y^{p - 1}$, since $p \nmid m$ and $p \nmid y$, we have:
- $p^{k + 2} \nmid p^{k + 1} m y^{p - 1}$
So:
- $p^{k + 2} \nmid x^p - y^p$
So by definition of $p$-adic valuation:
- $\map {\nu_p} {x^p - y^p} = k + 1$
$\blacksquare$
Sources
- July 1904: George David Birkhoff and Harry Schultz Vandiver: On the Integral Divisors of $a^n - b^n$ (Ann. Math. Vol. 5: pp. 173 – 180) www.jstor.org/stable/2007263: Theorem $\text{III}$.