Superelliptic equations arising from sums of consecutive powers
Volume 172 / 2016
Acta Arithmetica 172 (2016), 377-393
MSC: Primary 11D61; Secondary 11D41, 11F80, 11F11.
DOI: 10.4064/aa8305-12-2015
Published online: 18 February 2016
Abstract
Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^3+x^3+(x+1)^3=z^2$ (with $x, z \in \mathbb Z$). The generalization $(x-1)^k+x^k+(x+1)^k=z^n$ (with $x, z, n\in \mathbb Z$ and $n\ge 2$) was considered by Zhongfeng Zhang who solved it for $k \in \{ 2, 3, 4 \}$ using Frey–Hellegouarch curves and their corresponding Galois representations. In this paper, by employing some sophisticated refinements of this approach, we show that the only solutions for $k=5$ have $x=z=0$, 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.
