{"paper":{"title":"On a problem of Chen and Liu concerning the prime power factorization of $n!$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Johannes F. Morgenbesser, T. Stoll","submitted_at":"2011-10-21T15:03:45Z","abstract_excerpt":"For a fixed prime $p$, let $e_p(n!)$ denote the order of $p$ in the prime factorization of $n!$. Chen and Liu (2007) asked whether for any fixed $m$, one has $\\{e_p(n^2!) \\bmod m:\\; n\\in\\mathbb{Z}\\}=\\mathbb{Z}_m$ and $\\{e_p(q!) \\bmod m:\\; q {prime}\\}=\\mathbb{Z}_m$. We answer these two questions and show asymptotic formulas for $# \\{n<x: n \\equiv a \\bmod d,\\; e_p(n^2!)\\equiv r \\bmod m\\}$ and $# \\{q<x: q {prime}, q \\equiv a \\bmod d,\\; e_p(q!)\\equiv r \\bmod m\\}$. Furthermore, we show that for each $h\\geq 3$, we have $\\{n<x: n \\equiv a \\bmod d,\\; e_p(n^h!)\\equiv r \\bmod m\\} \\gg x^{4/(3h+1)}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1110.4814","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"}