On some Generalized Fermat Equations of the form $x^2+y^{2n} = z^p$

TL;DR

This paper investigates specific generalized Fermat equations, employing modularity and computational techniques to completely solve the case when p=7 and partially address other related equations.

Contribution

It provides a complete solution for the equation with p=7 and extends methods to solve a second related Fermat-type equation.

Findings

Complete resolution of x^2 + y^{2n} = z^{21} for coprime integers when p=7.

Asymptotic results for fixed prime p in the studied equations.

Solution of x^{2 extell{l}} + y^{2m} = z^{17} for primes extell{l,m} eq 5.

Abstract

The primary aim of this paper is to study the generalized Fermat equation \[ x^2+y^{2n} = z^{3p} \] in coprime integers $x$, $y$, and $z$, where $n \geq 2$ and $p$ is a fixed prime. Using modularity results over totally real fields and the explicit computation of Hilbert cuspidal eigenforms, we provide a complete resolution of this equation in the case $p=7$, and obtain an asymptotic result for fixed $p$. Additionally, using similar techniques, we solve a second equation, namely $x^{2\ell}+y^{2m} = z^{17}$, for primes $\ell,m \ne 5$.