{"paper":{"title":"Proof of a Conjecture on 6-colored Generalized Frobenius Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Liuquan Wang","submitted_at":"2015-07-11T11:50:31Z","abstract_excerpt":"Let $c\\phi_{k}(n)$ be the $k$-colored generalized Frobenius partition function. By employing the generating function of $c\\phi_{6}(3n+1)$ found by Hirschhorn, we prove that $c\\phi_{6}(27n+16)\\equiv 0$ (mod 243). This confirms a conjecture of E.X.W. Xia. We also find a congruence relation $c\\phi_{6}(81n+61) \\equiv 3 c\\phi_{6}(9n+7)$ (mod 243). Moreover, we show that $c\\phi_{6}(81n+61) \\equiv 0$ (mod 81), $c\\phi_{6}(243n+142) \\equiv 0$ (mod 243) and $c\\phi_{6}(729n+ 547) \\equiv 0$ (mod 243). We further conjecture that for $n\\ge 0$, $c\\phi_{6}(243n+142) \\equiv 0$ (mod 729)."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03101","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"}