On the generalized Fermat equation over totally real fields

TL;DR

This paper extends the asymptotic results of Fermat's Last Theorem to generalized Fermat equations over totally real fields, involving odd integer coefficients, broadening the scope of previous work.

Contribution

It generalizes prior asymptotic Fermat theorem results to equations with coefficients in totally real fields, including odd integers.

Findings

Proves asymptotic non-existence of solutions for generalized Fermat equations over totally real fields.

Extends techniques from Fermat's Last Theorem to broader classes of equations.

Provides new insights into the structure of solutions in algebraic number fields.

Abstract

In a recent paper, Freitas and Siksek proved an asypmtotic version of Fermat's Last Theorem for many totally real fields. We prove an extension of their result to generalized Fermat equations of the form $A x^p+B y^p+ C z^p=0$, where $A$, $B$, $C$ are odd integers belonging to a totally real field.