{"paper":{"title":"A note on $n!$ modulo $p$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"J. Hern\\'andez, M. Z. Garaev","submitted_at":"2015-05-21T21:44:41Z","abstract_excerpt":"Let $p$ be a prime, $\\varepsilon>0$ and $0<L+1<L+N < p$. We prove that if $p^{1/2+\\varepsilon}< N <p^{1-\\varepsilon}$, then $$ \\#\\{n!\\!\\!\\! \\pmod p;\\,\\, L+1\\le n\\le L+N\\} > c (N\\log N)^{1/2},\\,\\, c=c(\\varepsilon)>0. $$ We use this bound to show that any $\\lambda\\not\\equiv 0\\pmod p$ can be represented in the form $\\lambda \\equiv n_1!...n_7!\\pmod p$, where $n_i=o(p^{11/12})$. This slightly refines the previously known range for $n_i$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.05912","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"}