Solving Fermat-type equations x^5+y^5=dz^p

TL;DR

This paper introduces a new modularity-based method to solve Fermat-type equations of the form x^5 + y^5 = dz^p for positive integers d and primes p ≥ 7, improving previous results.

Contribution

The authors develop a novel approach leveraging modularity theorems to advance solutions for specific Fermat-type equations, surpassing earlier methods.

Findings

Enhanced solutions for x^5 + y^5 = dz^p equations with p ≥ 7

Method limitations analyzed for the case d=3

Improved results over previous work

Abstract

In this paper we are interested in solving the Fermat-type equations x^5+y^5=dz^p where d is a positive integer and p a prime number $\ge 7$. We describe a new method based on modularity theorems which allows us to improve all the results in a previous paper of the first author. We finally discuss the present limitations of the method by looking at the case d=3.