{"paper":{"title":"The least modulus for which consecutive polynomial values are distinct","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2013-04-22T15:39:13Z","abstract_excerpt":"Let $d\\ge4$ and $c\\in(-d,d)$ be relatively prime integers. We show that for any sufficiently large integer $n$ (in particular $n>24310$ suffices for $4\\le d\\le 36$), the smallest prime $p\\equiv c\\pmod d$ with $p\\ge(2dn-c)/(d-1)$ is the least positive integer $m$ with $2r(d)k(dk-c)\\ (k=1,\\ldots,n)$ pairwise distinct modulo $m$, where $r(d)$ is the radical of $d$. We also conjecture that for any integer $n>4$ the least positive integer $m$ such that $|\\{k(k-1)/2\\ \\mbox{mod}\\ m:\\ k=1,\\ldots,n\\}|= |\\{k(k-1)/2\\ \\mbox{mod}\\ m+2:\\ k=1,\\ldots,n\\}|=n$ is the least prime $p\\ge 2n-1$ with $p+2$ also prim"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.5988","kind":"arxiv","version":5},"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"}