{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2005:KNROGO4GYPCDNGZYVIUCBBFXSM","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":"3d525ae86222385657441663c3cda4f4706185c8bd4a2aeb5f890f868ce7a3ab","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-02-09T20:21:42Z","title_canon_sha256":"c0da663a0d350544ada797da8dc139d4ff81b0708fd0e85f59a7c40e42577b7f"},"schema_version":"1.0","source":{"id":"math/0502187","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"math/0502187","created_at":"2026-07-04T15:03:27Z"},{"alias_kind":"arxiv_version","alias_value":"math/0502187v4","created_at":"2026-07-04T15:03:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0502187","created_at":"2026-07-04T15:03:27Z"},{"alias_kind":"pith_short_12","alias_value":"KNROGO4GYPCD","created_at":"2026-07-04T15:03:27Z"},{"alias_kind":"pith_short_16","alias_value":"KNROGO4GYPCDNGZY","created_at":"2026-07-04T15:03:27Z"},{"alias_kind":"pith_short_8","alias_value":"KNROGO4G","created_at":"2026-07-04T15:03:27Z"}],"graph_snapshots":[{"event_id":"sha256:7ed0ae4a8e51c9da2974ceb7bdd45078f5e5203e128260f2be2dd00f51418e93","target":"graph","created_at":"2026-07-04T15:03:27Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/math/0502187/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let q>1 and m>0 be relatively prime integers. We find an explicit period $\\nu_m(q)$ such that for any integers n>0 and r we have $[n+\\nu_m(q),r]_m(a)=[n,r]_m(a) (mod q)$ whenever a is an integer with $\\gcd(1-(-a)^m,q)=1$, or a=-1 (mod q), or a=1 (mod q) and 2|m, where $[n,r]_m(a)=\\sum_{k=r(mod m)}\\binom{n}{k}a^k$. This is a further extension of a congruence of Glaisher.","authors_text":"Roberto Tauraso, Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"","primary_cat":"math.NT","submitted_at":"2005-02-09T20:21:42Z","title":"Congruences for sums of binomial coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0502187","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":"sha256:5d1b4b97ff5ce89a8a52a41f134e5ac14983d85c32392c26b83d6f833f477a09","target":"record","created_at":"2026-07-04T15:03:27Z","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":"3d525ae86222385657441663c3cda4f4706185c8bd4a2aeb5f890f868ce7a3ab","cross_cats_sorted":["math.CO"],"license":"","primary_cat":"math.NT","submitted_at":"2005-02-09T20:21:42Z","title_canon_sha256":"c0da663a0d350544ada797da8dc139d4ff81b0708fd0e85f59a7c40e42577b7f"},"schema_version":"1.0","source":{"id":"math/0502187","kind":"arxiv","version":4}},"canonical_sha256":"5362e33b86c3c4369b38aa282084b79319f2314ee4c425ac519e50e43c1fad72","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5362e33b86c3c4369b38aa282084b79319f2314ee4c425ac519e50e43c1fad72","first_computed_at":"2026-07-04T15:03:27.624201Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-04T15:03:27.624201Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"S+Zl1iVw/FQUgZoleAEABsPzfyLV2qTBHPBpcjPQCACFK21vTEh9SFNByCtcmjA/Tb6sQ1yXjPDOldAQ/tTEAA==","signature_status":"signed_v1","signed_at":"2026-07-04T15:03:27.624618Z","signed_message":"canonical_sha256_bytes"},"source_id":"math/0502187","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d1b4b97ff5ce89a8a52a41f134e5ac14983d85c32392c26b83d6f833f477a09","sha256:7ed0ae4a8e51c9da2974ceb7bdd45078f5e5203e128260f2be2dd00f51418e93"],"state_sha256":"3093837a97e444f0d60438867555a9f5c98284aa8fb508098bdb7609ba6dea1c"}