{"paper":{"title":"Combinatorial congruences modulo prime powers","license":"","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Donald M. Davis, Zhi-Wei Sun","submitted_at":"2005-08-04T07:28:54Z","abstract_excerpt":"Let p be any prime, and let a and n be nonnegative integers. Let $r\\in Z$ and $f(x)\\in Z[x]$. We establish the congruence\n  $$p^{\\deg f}\\sum_{k=r(mod p^a)}\\binom{n}{k}(-1)^k f((k-r)/p^a) =0 (mod p^{\\sum_{i=a}^{\\infty}[n/p^i]})$$ (motivated by a conjecture arising from algebraic topology), and obtain the following vast generalization of Lucas' theorem: If a is greater than one, and $l,s,t$ are nonnegative integers with $s,t<p$, then\n  $$\\frac{1}{[n/p^{a-1}]!} \\sum_{k=r(mod p^a)} \\binom{pn+s}{pk+t}(-1)^{pk}((k-r)/p^{a-1})^l =\\frac {1}{[n/p^{a-1}]!} \\sum_{k=r(mod p^a)}\\binom{n}{k}\\binom{s}{t}(-1)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0508087","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/math/0508087/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}