{"paper":{"title":"Congruences modulo powers of 3 for 2-color partition triples","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dazhao Tang","submitted_at":"2018-05-23T03:07:21Z","abstract_excerpt":"Let $p_{k,3}(n)$ enumerate the number of 2-color partition triples of $n$ where one of the colors appears only in parts that are multiples of $k$. In this paper, we prove several infinite families of congruences modulo powers of 3 for $p_{k,3}(n)$ with $k=1, 3$, and $9$. For example, for all integers $n\\geq0$ and $\\alpha\\geq1$, we prove that \\begin{align*} p_{3,3}\\left(3^{\\alpha}n+\\dfrac{3^{\\alpha}+1}{2}\\right) &\\equiv0\\pmod{3^{\\alpha+1}} \\end{align*} and \\begin{align*} p_{3,3}\\left(3^{\\alpha+1}n+\\dfrac{5\\times3^{\\alpha}+1}{2}\\right) &\\equiv0\\pmod{3^{\\alpha+4}}. \\end{align*}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.08942","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"}