{"paper":{"title":"Structure and asymptotics for Catalan numbers modulo primes using automata","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Rob Burns","submitted_at":"2017-01-11T14:03:43Z","abstract_excerpt":"Let $C_n$ be the $n$th Catalan number. We show that the asymptotic density of the set $\\{n: C_n \\equiv 0 \\mod p \\}$ is $1$ for all primes $p$, We also show that if $n = p^k -1$ then $C_n \\equiv -1 \\mod p$. Finally we show that if $n \\equiv \\{ \\frac{p+1}{2}, \\frac{p+3}{2}, ..., p-2 \\} \\mod p$ then $p$ divides $C_n$. All results are obtained using the automata method of Rowland and Yassawi."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.02975","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"}