{"paper":{"title":"Sylvester's Theorem and the Non-integrality of a certain Binomial Sum","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Daniel L\\'opez-Aguayo, Florian Luca","submitted_at":"2015-07-30T18:34:21Z","abstract_excerpt":"In this note, we show that $S(n,r):=\\sum_{k=0}^{n} \\binom{n}{k}\\frac{k}{k+r}$ is not an integer for any positive integer $n$ and $r\\in \\{1,2,3,4,5,6\\}$ and for $n\\le r-1$. This gives a partial answer to a conjecture of [3]."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.02927","kind":"arxiv","version":3},"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"}