{"paper":{"title":"Progress towards a nonintegrality conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Carl Pomerance, Daniel L\\'opez-Aguayo, Shanta Laishram, Thotsaphon Thongjunthug","submitted_at":"2019-03-19T15:09:52Z","abstract_excerpt":"Given $r \\in \\mathbb{N}$, define the function $S_{r}: \\mathbb{N} \\rightarrow \\mathbb{Q}$ by $S_{r}(n)=\\displaystyle \\sum_{k=0}^{n} \\frac{k}{k+r} \\binom{n}{k}$. In $2015$, the second author conjectured that there are infinitely many $r \\in \\mathbb{N}$ such that $S_{r}(n)$ is nonintegral for all $n \\geq 1$, and proved that $S_{r}(n)$ is not an integer for $r \\in \\{2,3,4\\}$ and for all $n \\geq 1$. In $2016$, Florian Luca and the second author raised the stronger conjecture that for any $r \\geq 1$, $S_{r}(n)$ is nonintegral for all $n \\geq 1$. They proved that $S_{r}(n)$ is nonintegral for $r \\in "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.08043","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"}