{"paper":{"title":"Forbidden integer ratios of consecutive power sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ioulia N. Baoulina, Pieter Moree","submitted_at":"2015-10-20T21:16:20Z","abstract_excerpt":"Let $S_k(m):=1^k+2^k+\\cdots+(m-1)^k$ denote a power sum. In 2011 Bernd Kellner formulated the conjecture that for $m\\ge 4$ the ratio $S_k(m+1)/S_k(m)$ of two consecutive power sums is never an integer. We will develop some techniques that allow one to exclude many integers $\\rho$ as a ratio and combine them to exclude the integers $3\\le \\rho\\le 1501$ and, assuming a conjecture on irregular primes to be true, a set of density $1$ of ratios $\\rho$. To exclude a ratio $\\rho$ one has to show that the Erd\\H{o}s-Moser type equation $(\\rho-1)S_k(m)=m^k$ has no non-trivial solutions."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.06064","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"}