Computing rational solutions to $Ax^p+By^p+Cz^p=0$

TL;DR

This survey explores methods for finding rational solutions to the equation $Ax^p+By^p+Cz^p=0$ by analyzing associated hyperelliptic curves and employing computational tools to identify rational points.

Contribution

It introduces a computational approach to study rational solutions of the equation through hyperelliptic curves and their rational points.

Findings

Hyperelliptic curve associated with the equation is analyzed.

Computational tools are used to find rational points on the curve.

The approach provides insights into the existence of rational solutions.

Abstract

In this survey, we studied the possibility of finding rational solutions to the equation $Ax^p+By^p+Cz^p=0$ via its attached hyperelliptic curve $Y^2=X^p+A^2(BC)^{p-1}/4$ and its rational points computed using computational tools.