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

TL;DR

This work studies rational solutions to $Ax^p+By^p+Cz^p=0$ by associating a hyperelliptic curve $Y^2 = X^p + \dfrac{A^2(BC)^{p-1}}{4}$ and leveraging rational points on this curve. The approach relies on a standard variable change, the genus–rank relation $p=2g+1$, and finiteness results from Faltings, with practical point-finding enabled by Magma and Chabauty-Coleman bounds when the Jacobian rank is small. The method requires careful handling of the ordering of $A,B,C$ and degenerate cases, and it reconstructs potential solutions $(x,y,z)$ only from $E(\mathbb{Q})$ points with $XY\neq 0$, via $p$-th roots of certain rational expressions. Through explicit examples, the paper demonstrates how curve rank and rational points determine whether nontrivial rational solutions exist, illustrating a computational pathway for generalized Fermat-type Diophantine equations over $\mathbb{Q}$.

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.

Theorems & Definitions (6)

• Theorem 3.1
• Theorem 3.2
• Remark 4.1
• Example 1
• Example 2
• Example 3