The equation $(x-d)^5 + x^5 + (x+d)^5 = y^n$

TL;DR

This paper solves a specific exponential Diophantine equation involving fifth powers under certain coprimality conditions, using advanced number theory techniques like modular forms and Chabauty methods.

Contribution

It generalizes previous results by solving the equation with gcd conditions and applies modern tools such as Frey-Hellegouarch curves and Chabauty techniques.

Findings

Solved the equation $(x-d)^5 + x^5 + (x+d)^5 = y^n$ under gcd conditions.

Extended earlier work by incorporating new number theory methods.

Demonstrated the effectiveness of modular forms and Chabauty techniques in solving such equations.

Abstract

In this paper, we solve the equation of the title under the assumption that $\gcd(x,d)=1$ and $n\geq 2$. This generalizes earlier work of the first author, Patel and Siksek [BPS16]. Our main tools include Frey-Hellegouarch curves and associated modular forms, and an assortment of Chabauty-type techniques for determining rational points on curves of small positive genus.