{"paper":{"title":"A short Proof of a conjecture by Hirschhorn and Sellers on Overpartitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2014-07-21T09:30:29Z","abstract_excerpt":"Let $\\overline{p}(n)$ be the number of overpartitions of $n$, we establish and give a short elementary proof of the following congruence \\[\\overline{p}({{4}^{\\alpha }}(40n+35))\\equiv 0 \\, (\\bmod \\, 40),\\] where $\\alpha ,n $ are nonnegative integers. By letting $\\alpha =0$ we proved a conjecture of Hirschhorn and Sellers. Some new congruences for $\\overline{p}(n)$ modulo 3 and 5 have also been found, including the following two infinite families of Ramanujan-type congruences: for any integers $n\\ge 0$ and $\\alpha \\ge 1$, \\[\\overline{p}({{5}^{2\\alpha +1}}(5n+1))\\equiv \\overline{p}({{5}^{2\\alpha "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.5430","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"}