On the sum of fourth powers in arithmetic progression

TL;DR

This paper proves that the sum of four specific fourth powers in an arithmetic progression cannot equal a perfect power for coprime integers, using advanced modular techniques involving Frey curves.

Contribution

It introduces a novel modular approach with Frey curves over quadratic fields to solve a classical exponential Diophantine equation.

Findings

No solutions for the equation with coprime x, y for all n > 1

Utilizes Frey curves over Q(√30)

Advances methods in exponential Diophantine equations

Abstract

We prove that the equation ${ (x - y)^4 + x^4 + (x + y)^4 = z^n }$ has no integer solutions ${ x, y, z}$ with ${ \gcd(x, y) = 1 }$ for all integers ${ n > 1 }$. We mainly use a modular approach with two Frey ${ \mathbb{Q} }$-curves defined over the field ${ \mathbb{Q}( \sqrt{30} ) }$.