{"paper":{"title":"The 3-adic valuations of Stirling numbers of the first kind","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Long Chen, Min Qiu, Zongbing Lin","submitted_at":"2021-09-28T03:16:51Z","abstract_excerpt":"Let $v_3$ denote the usual $3$-adic valuation, and let $s(n, k)$ be the unsigned Stirling number of the first kind. In this paper, for $a\\in\\{1,2\\}$, we determine the values of $v_3(s(a3^n, k))$ for all $1\\le k\\le a3^n$. More precisely, for each admissible pair $(m, k)$, we obtain an explicit formula for $v_3(s(a3^n, a3^m-k))$. The proof combines properties of the $m$-th Stirling numbers of the first kind with a detailed analysis of the relevant $3$-adic orders. As a consequence, we prove the case $p=3$ of a conjecture of Hong and Qiu proposed in 2020. We also derive formulas near the diagonal"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2109.13458","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2109.13458/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}