{"paper":{"title":"Two new kinds of numbers and related divisibility results","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2014-08-21T16:15:50Z","abstract_excerpt":"We mainly introduce two new kinds of numbers given by $$R_n=\\sum_{k=0}^n\\binom nk\\binom{n+k}k\\frac1{2k-1}\\quad\\ (n=0,1,2,...)$$ and $$S_n=\\sum_{k=0}^n\\binom nk^2\\binom{2k}k(2k+1)\\quad\\ (n=0,1,2,...).$$ We find that such numbers have many interesting arithmetic properties. For example, if $p\\equiv1\\pmod 4$ is a prime with $p=x^2+y^2$ (where $x\\equiv1\\pmod 4$ and $y\\equiv0\\pmod 2$), then $$R_{(p-1)/2}\\equiv p-(-1)^{(p-1)/4}2x\\pmod{p^2}.$$ Also, $$\\frac1{n^2}\\sum_{k=0}^{n-1}S_k\\in\\mathbb Z\\ \\ {and}\\ \\ \\frac1n\\sum_{k=0}^{n-1}S_k(x)\\in\\mathbb Z[x]\\quad\\text{for all}\\ n=1,2,3,...,$$ where $S_k(x)=\\s"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1408.5381","kind":"arxiv","version":10},"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"}