On the generalized Fermat equation of signature $(5,p,3)$

TL;DR

This paper investigates solutions to the generalized Fermat equation with signature (5,p,3) using hypergeometric motives and the modular method, providing new insights into ramification, trivial solutions, and obstructions to the method's success.

Contribution

It offers an explicit ramification analysis, identifies obstructions to the modular method, and proves nonexistence of certain solutions assuming a large image conjecture.

Findings

No nontrivial primitive solutions for large primes p when q=5, r=3, and 3 does not divide c.

Explicit description of ramification behavior at primes dividing 2qr.

Identification of a general obstruction to the modular method's applicability.

Abstract

In this article we study solutions to the generalized Fermat equation $x^q+y^p+z^r=0 $ using hypergeometric motives within the framework of the modular method. In doing so, we give an explicit description of the ramification behavior at primes dividing $2qr$ and analyze the contribution of trivial solutions. We identify a general obstruction to the modular method that accounts for its failure in many instances. As an application, assuming a standard large image conjecture, we prove that the previous equation admits no nontrivial primitive solutions $(a,b,c)$ with $3 \nmid c$, when $q=5,$ $r=3$ and $p$ is a prime sufficiently large.