{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:CF4RIQSZLNBVE47H4RUPULTXWT","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"6a374073569a0163ba4f8468f9ed267576e5a414d64d2e4f5afec643780b7d18","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-07-01T19:32:16Z","title_canon_sha256":"2ff877af5aeb1328f3e85a14521c61c409c88733f4ca15b233586b42faa199c5"},"schema_version":"1.0","source":{"id":"math/0507008","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0507008","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"arxiv_version","alias_value":"math/0507008v4","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0507008","created_at":"2026-05-18T01:38:24Z"},{"alias_kind":"pith_short_12","alias_value":"CF4RIQSZLNBV","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_16","alias_value":"CF4RIQSZLNBVE47H","created_at":"2026-05-18T12:25:53Z"},{"alias_kind":"pith_short_8","alias_value":"CF4RIQSZ","created_at":"2026-05-18T12:25:53Z"}],"graph_snapshots":[{"event_id":"sha256:a1c557fce4f9cf02d2201cf9a54778a21c57571638f390bb7cfc6b64decd2479","target":"graph","created_at":"2026-05-18T01:38:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $a\\equiv-1(mod p)$.","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2005-07-01T19:32:16Z","title":"Polynomial extension of Fleck's congruence"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0507008","kind":"arxiv","version":4},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:ba61647a4f2215154ef19ebc3081f0c1e3c12c5481e9f81f3d1db53b15c6beba","target":"record","created_at":"2026-05-18T01:38:24Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"6a374073569a0163ba4f8468f9ed267576e5a414d64d2e4f5afec643780b7d18","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-07-01T19:32:16Z","title_canon_sha256":"2ff877af5aeb1328f3e85a14521c61c409c88733f4ca15b233586b42faa199c5"},"schema_version":"1.0","source":{"id":"math/0507008","kind":"arxiv","version":4}},"canonical_sha256":"11791442595b435273e7e468fa2e77b4d976b2362c08953fba8c86ddf60adc73","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"11791442595b435273e7e468fa2e77b4d976b2362c08953fba8c86ddf60adc73","first_computed_at":"2026-05-18T01:38:24.945284Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:38:24.945284Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1C+PLZ6QFC2KDIepTvSiJVKahUMItHTq3BS9N/xtI5z3nm0PlivwzTgZOpkKBEHvrrZZhEBQp8RnVP7w/QdWCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T01:38:24.946008Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0507008","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ba61647a4f2215154ef19ebc3081f0c1e3c12c5481e9f81f3d1db53b15c6beba","sha256:a1c557fce4f9cf02d2201cf9a54778a21c57571638f390bb7cfc6b64decd2479"],"state_sha256":"c8928a40eccc24e70ac2110b1733e246e31cca87d183500d40953f78efb29b4f"}