A p-adic identity for Wieferich primes

Abstract

Let $n$ be a positive integer, $p$ be an odd prime and integers $a,b\ne 0$ with $gcd(a,b)=1$, $p\nmid ab$, and $p|(a^n\pm b^n)$, we prove the identity $${\nu }_p(a^n\pm b^n)-{\nu }_p(n)={\nu }_p(a^{p-1}-b^{p-1}).$$ An unintended interesting immediate consequence is the following variant of Wieferich's criterion for FLT : Let $x^n+y^n=z^n$ with $n$ prime and $x,y,z$ pairwise relatively prime. Then every odd prime $p|y$ satisfies ${\nu }_p(z^{p-1}-x^{p-1})\ge n-1$ and every odd prime $p|x$ satisfies ${\nu }_p(z^{p-1}-y^{p-1})\ge n-1$, and every odd prime $p|z$ satisfies ${\nu }_p(x^{p-1}-y^{p-1})\ge n-1$, ie. every odd prime dividing $xyz$ is a Wieferich prime of order at least $n-1$ to some base pair. In the "first case" where $n\nmid xyz$, the lower bound for the Wieferich order can be improved to $n$. This gives us very strong intuition why there should not be any solution even for moderately large $n$.

5 pages