Superelliptic equations arising from sums of consecutive powers

2015/09/22 by Michael A. Bennett, Vandita Patel, Bennett, Michael A. +3
Mathematics · #11D61 #FOS: Mathematics #Number Theory (math.NT) #math.NT #msc:11D61

paper · pdf · doi:10.48550/arxiv.1509.06619

arxiv created 2015/09/22 · arxiv updated 2015/09/23

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+x^k+(x+1)^k=zⁿ (with x, z, n integers and n ≥ 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.

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