The Generalized Fermat Equation $x^2 + y^3 = z^{25}$

TL;DR

This paper investigates the solutions to the generalized Fermat equation x^2 + y^3 = z^{25} by reducing it to five specific equations involving homogeneous forms over sextic number fields, building on known parameterizations.

Contribution

It extends the analysis of Fermat-type equations by reducing the problem to solving five specialized equations over sextic fields, leveraging existing parameterizations.

Findings

Reduced the original equation to five equations of form H(u,v)=w^5

Connected solutions to known parameterizations of simpler equations

Provided a framework for solving similar high-degree equations

Abstract

We consider the generalized Fermat equation (*) $x^2 + y^3 = z^{25}$. Using the known parameterization of the primitive integral solutions to $x^2 + y^3 = z^5$ (due to Edwards), we reduce the solution of (*) to the solution of five specific equations of the form $H(u,v) = w^5$, where $H$ is homogeneous of degree $10$ with coefficients in a sextic number field $K$, $u$ and $v$ are coprime (rational) integers, and $w \in K$.