# Asymptotic Generalized Fermat's Last Theorem over Number Fields

## TL;DR

This paper extends the asymptotic version of Fermat's Last Theorem to many imaginary quadratic number fields, building on previous work for totally real fields and generalized equations.

## Contribution

It generalizes existing asymptotic Fermat theorems to include many imaginary quadratic fields, using combined techniques from prior research.

## Key findings

- Proves asymptotic generalized Fermat's Last Theorem for many imaginary quadratic fields.
- Extends previous results from totally real to imaginary quadratic fields.
- Utilizes combined methods from prior works to achieve the extension.

## Abstract

Recent work of Freitas and Siksek showed that an asymptotic version of Fermat's Last Theorem holds for many totally real fields. Later this result was extended by Deconinck to generalized Fermat equations of the form $Ax^p +By^p +Cz^p = 0$, where A;B;C are odd integers belonging to a totally real field. Another extension was given by Sengun and Siksek. They showed that the Fermat equation holds asymptotically for imaginary quadratic number fields satisfying usual conjectures about modularity. In this work, combining their techniques we extend their results about the generalized Fermat equations to imaginary quadratic fields. More specifically we prove that the asymptotic generalized Fermat's Last Theorem holds for many quadratic imaginary number fields.