{"paper":{"title":"Congruences Modulo Powers of 3 for 3- and 9-Colored Generalized Frobenius Partitions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Liuquan Wang","submitted_at":"2018-01-24T12:12:38Z","abstract_excerpt":"Let $c\\phi_{k}(n)$ be the number of $k$-colored generalized Frobenius partitions of $n$. We establish some infinite families of congruences for $c\\phi_{3}(n)$ and $c\\phi_{9}(n)$ modulo arbitrary powers of 3, which refine the results of Kolitsch. For example, for $k\\ge 3$ and $n\\ge 0$, we prove that \\[c\\phi_{3}\\Big(3^{2k}n+\\frac{7\\cdot 3^{2k}+1}{8}\\Big) \\equiv 0 \\pmod{3^{4k+5}}.\\] We give two different proofs to the congruences satisfied by $c\\phi_{9}(n)$. One of the proofs uses an relation between $c\\phi_{9}(n)$ and $c\\phi_{3}(n)$ due to Kolitsch, for which we provide a new proof in this paper"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1801.07949","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"}