On the solutions to $Ax^p+By^p+Cz^p=0$ over quadratic fields

TL;DR

This paper investigates solutions to the equation $Ax^p+By^p+Cz^p=0$ over quadratic fields, establishing conditions for solutions and their rationality, and proving non-existence in certain cases.

Contribution

It provides necessary conditions for solutions over quadratic fields and characterizes when solutions are rational or lie outside the rationals.

Findings

Solutions exist under specific conditions on $A,B,C$ and the field.

Solutions are rational for coprime integers $A,B,C$ under certain conditions.

No solutions exist outside $Q$ when $BC eq pm 1$.

Abstract

We provide the necessary conditions for the existence of solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ over any quadratic number field $K$ with $A,B,C$ pth powerfree integer numbers. We determine when $x$, $y$ and $z$ are rational numbers for pairwise coprime integers $A$, $B$ and $C$. Moreover, we prove that $x$, $y$ and $z$ are in $K\setminus\mathbb{Q}$ when $BC=\pm 1$ and $A\neq \pm 2$. Finally, we prove that no solutions $(x,y,z)$ to $Ax^p+By^p+Cz^p=0$ exist in $K\setminus\mathbb{Q}$ when $BC\neq \pm 1$.