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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 $a\\equiv-1(mod p)$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"math/0507008","kind":"arxiv","version":4},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2005-07-01T19:32:16Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"2ff877af5aeb1328f3e85a14521c61c409c88733f4ca15b233586b42faa199c5","abstract_canon_sha256":"6a374073569a0163ba4f8468f9ed267576e5a414d64d2e4f5afec643780b7d18"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:38:24.946008Z","signature_b64":"1C+PLZ6QFC2KDIepTvSiJVKahUMItHTq3BS9N/xtI5z3nm0PlivwzTgZOpkKBEHvrrZZhEBQp8RnVP7w/QdWCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"11791442595b435273e7e468fa2e77b4d976b2362c08953fba8c86ddf60adc73","last_reissued_at":"2026-05-18T01:38:24.945284Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:38:24.945284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"math/0507008","created_at":"2026-05-18T01:38:24.945419+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0507008v4","created_at":"2026-05-18T01:38:24.945419+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0507008","created_at":"2026-05-18T01:38:24.945419+00:00"},{"alias_kind":"pith_short_12","alias_value":"CF4RIQSZLNBV","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_16","alias_value":"CF4RIQSZLNBVE47H","created_at":"2026-05-18T12:25:53.335082+00:00"},{"alias_kind":"pith_short_8","alias_value":"CF4RIQSZ","created_at":"2026-05-18T12:25:53.335082+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT","json":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT.json","graph_json":"https://pith.science/api/pith-number/CF4RIQSZLNBVE47H4RUPULTXWT/graph.json","events_json":"https://pith.science/api/pith-number/CF4RIQSZLNBVE47H4RUPULTXWT/events.json","paper":"https://pith.science/paper/CF4RIQSZ"},"agent_actions":{"view_html":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT","download_json":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT.json","view_paper":"https://pith.science/paper/CF4RIQSZ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0507008&json=true","fetch_graph":"https://pith.science/api/pith-number/CF4RIQSZLNBVE47H4RUPULTXWT/graph.json","fetch_events":"https://pith.science/api/pith-number/CF4RIQSZLNBVE47H4RUPULTXWT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT/action/storage_attestation","attest_author":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT/action/author_attestation","sign_citation":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT/action/citation_signature","submit_replication":"https://pith.science/pith/CF4RIQSZLNBVE47H4RUPULTXWT/action/replication_record"}},"created_at":"2026-05-18T01:38:24.945419+00:00","updated_at":"2026-05-18T01:38:24.945419+00:00"}