{"paper":{"title":"On Fleck quotients","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Daqing Wan, Zhi-Wei Sun","submitted_at":"2006-03-20T19:18:00Z","abstract_excerpt":"Let $p$ be a prime, and let $n>0$ and $r$ be integers. In this paper we study Fleck's quotient $$F_p(n,r)=(-p)^{-\\lfloor(n-1)/(p-1)\\rfloor} \\sum_{k=r(mod p)}\\binom {n}{k}(-1)^k\\in Z.$$\n  We determine $F_p(n,r)$ mod $p$ completely by certain number-theoretic and combinatorial methods; consequently, if $2\\le n\\le p$ then $$\\sum_{k=1}^n(-1)^{pk-1}\\binom{pn-1}{pk-1} \\equiv(n-1)!B_{p-n}p^n (mod p^{n+1}),$$ where $B_0,B_1,...$ are Bernoulli numbers. We also establish the Kummer-type congruence $F_p(n+p^a(p-1),r)\\equiv F_p(n,r) (mod p^a)$ for $a=1,2,3,...$, and reveal some connections between Fleck's"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0603462","kind":"arxiv","version":3},"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"}