Superelliptic equations arising from sums of consecutive powers

TL;DR

This paper investigates superelliptic equations derived from sums of consecutive powers, extending previous work by solving for specific exponents using advanced number-theoretic techniques and computational refinements.

Contribution

It proves that the only solution for the case k=5 is trivial and that no solutions exist for k=6, employing refined Frey-Hellegouarch curve methods with computational innovations.

Findings

Only trivial solution for k=5: x=z=0

No solutions for k=6

Refined approach reduces computational complexity

Abstract

Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^3+x^3+(x+1)^3=z^2$ in integers $x$, $z$. The generalization $(x-1)^k+x^k+(x+1)^k=z^n$ (with $x$, $z$, $n$ integers and $n \ge 2$) was considered by Zhongfeng Zhang who solved it for $k=2$, $3$, $4$ using Frey-Hellegouarch curves and their Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solution for $k=5$ is $x=z=0$, and that there are no solutions for $k=6$. The chief innovation we employ is a computational one, which enables us to avoid the full computation of data about cuspidal newforms of high level.