A p-adic identity for Wieferich primes

TL;DR

This paper establishes a p-adic identity relating valuations of sums of powers and applies it to derive new criteria for Wieferich primes, providing insights into the non-existence of solutions to certain Fermat equations.

Contribution

The paper introduces a novel p-adic identity connecting valuations of a^n ± b^n and p-1 powers, with implications for Wieferich primes and Fermat's Last Theorem.

Findings

Derived a p-adic identity for valuations of power sums.

Established new bounds for Wieferich primes related to Fermat solutions.

Provided evidence supporting the non-existence of solutions for large exponents.

Abstract

Let $n$ be a positive integer, $p$ be an odd prime and integers $a,b \not= 0$ 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 $n$.