{"paper":{"title":"Polynomial extension of Fleck's congruence","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2005-07-01T19:32:16Z","abstract_excerpt":"Let $p$ be a prime, and let $f(x)$ be an integer-valued polynomial. By a combinatorial approach, we obtain a nontrivial lower bound of the $p$-adic order of the sum $$\\sum_{k=r(mod p^{\\beta})}\\binom{n}{k}(-1)^k f([(k-r)/p^{\\alpha}]),$$ where $\\alpha\\ge\\beta\\ge 0$, $n\\ge p^{\\alpha-1}$ and $r\\in Z$. This polynomial extension of Fleck's congruence has various backgrounds and several consequences such as $$\\sum_{k=r(mod p^\\alpha)}\\binom{n}{k} a^k\\equiv 0 (mod p^{[(n-p^{\\alpha-1})/\\phi(p^\\alpha)]})$$ provided that $\\alpha>1$ and $a\\equiv-1(mod p)$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0507008","kind":"arxiv","version":4},"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"}