Bennett, Michael A., Patel, Vandita and Siksek, Samir (2016) Superelliptic equations arising from sums of consecutive powers. Acta Arithmetica, 172 . pp. 377-393. doi:10.4064/aa8305-12-2015 ISSN 0065-1036.

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Abstract

Using only elementary arguments, Cassels solved the Diophantine equation (x−1)3+x3+(x+1)3=z2 in integers x, z. The generalization (x−1)k+xk+(x+1)k=zn (with x, z, n integers and n≥2) was considered by Zhongfeng Zhang who solved it for k E {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.

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Elliptic functions
Journal or Publication Title:	Acta Arithmetica
Publisher:	Polska Akademia Nauk: Instytut Matematyczny
ISSN:	0065-1036
Official Date:	18 February 2016
Dates:	Date Event 18 February 2016 Published 17 December 2015 Accepted
Volume:	172
Page Range:	pp. 377-393
DOI:	10.4064/aa8305-12-2015
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Funder:	Natural Sciences and Engineering Research Council of Canada (NSERC), Engineering and Physical Sciences Research Council (EPSRC)
Grant number:	EP/K034383/1 (EPSRC)
Open Access Version:	ArXiv

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