Superelliptic equations arising from sums of consecutive powers

Michael A. Bennett¹ ; Vandita Patel² ; Samir Siksek²

¹Department of Mathematics University of British Columbia Vancouver, BC, V6T 1Z2 Canada
²Mathematics Institute University of Warwick Coventry CV4 7AL, United Kingdom

Acta Arithmetica, Tome 172 (2016) no. 4, pp. 377-393

Résumé

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.

Détail

DOI : 10.4064/aa8305-12-2015

Keywords: using only elementary arguments cassels solved diophantine equation x mathbb generalization x mathbb considered zhongfeng zhang who solved using frey hellegouarch curves their corresponding galois representations paper employing sophisticated refinements approach only solutions have there solutions chief innovation employ computational which enables avoid full computation about cuspidal newforms high level