Using only elementary arguments, Cassels solved the Diophantine equation (x-1)³ + x³ + (x+1)³ = z² (with x, z ∈ ℤ). The generalization (with x, z, n ∈ ℤ and n ≥ 2) was considered by Zhongfeng Zhang who solved it for k ∈ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-aa8305-12-2015,
author = {Michael A. Bennett and Vandita Patel and Samir Siksek},
title = {Superelliptic equations arising from sums of consecutive powers},
journal = {Acta Arithmetica},
volume = {172},
year = {2016},
pages = {377-393},
zbl = {06574968},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8305-12-2015}
}
Michael A. Bennett; Vandita Patel; Samir Siksek. Superelliptic equations arising from sums of consecutive powers. Acta Arithmetica, Tome 172 (2016) pp. 377-393. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-aa8305-12-2015/