Beyond and slightly aside of the Fermat Desert

TL;DR

This paper provides a cyclotomic proof that certain equations involving prime exponents have no solutions in coprime integers, thereby supporting Fermat's Last Theorem.

Contribution

It introduces a cyclotomic approach to prove the non-existence of solutions for a specific class of equations related to Fermat's Last Theorem.

Findings

Proves the equation has no solutions for prime p > 3 in coprime integers.

Establishes a cyclotomic proof technique for related exponential equations.

Supports Fermat's Last Theorem through this specific case.

Abstract

We give a cyclotomic proof of the fact that the equation $\frac{x^p + y^p}{x+y} = p^e z^p$ has no solutions in coprime integers $x,y,z$ and $p > 3$, a prime. This implies in particular Fermat's Last Theorem.