{"paper":{"title":"Superelliptic equations arising from sums of consecutive powers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Michael A. Bennett, Samir Siksek, Vandita Patel","submitted_at":"2015-09-22T14:32:35Z","abstract_excerpt":"Using only elementary arguments, Cassels solved the Diophantine equation $(x-1)^3+x^3+(x+1)^3=z^2$ in integers $x$, $z$. The generalization $(x-1)^k+x^k+(x+1)^k=z^n$ (with $x$, $z$, $n$ integers and $n \\ge 2$) 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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1509.06619","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}