{"paper":{"title":"Power values of sums of certain products of consecutive integers and related results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maciej Ulas, Szabolcs Tengely","submitted_at":"2018-09-12T08:24:01Z","abstract_excerpt":"Let $n$ be a non-negative integer and put $p_{n}(x)=\\prod_{i=0}^{n}(x+i)$. In the first part of the paper, for given $n$, we study the existence of integer solutions of the Diophantine equation $$ y^m=p_{n}(x)+\\sum_{i=1}^{k}p_{a_{i}}(x), $$ where $m\\in\\N_{\\geq 2}$ and $a_{1}<a_{2}<\\ldots <a_{k}<n$. This equation can be considered as a generalization of the Erd\\H{o}s-Selfridge Diophantine equation $y^m=p_{n}(x)$. We present some general finiteness results concerning the integer solutions of the above equation. In particular, if $n\\geq 2$ with $a_{1}\\geq 2$, then our equation has only finitely m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.04304","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"}