On the generalized Fermat equation over totally real 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.