{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:3NOHSQZV6TVUSZT7JV3FZKHEWE","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":"87f815efdcf17a08d81ee20f7308a1485ffa3912919a7ea0bde793f4eed80632","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-24T13:47:04Z","title_canon_sha256":"8d337bc93e67bb4d23a43411b0307bb6d2f99cc656c7f4076d9fe84c4123df2b"},"schema_version":"1.0","source":{"id":"1108.4840","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1108.4840","created_at":"2026-05-18T04:14:46Z"},{"alias_kind":"arxiv_version","alias_value":"1108.4840v1","created_at":"2026-05-18T04:14:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.4840","created_at":"2026-05-18T04:14:46Z"},{"alias_kind":"pith_short_12","alias_value":"3NOHSQZV6TVU","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_16","alias_value":"3NOHSQZV6TVUSZT7","created_at":"2026-05-18T12:26:18Z"},{"alias_kind":"pith_short_8","alias_value":"3NOHSQZV","created_at":"2026-05-18T12:26:18Z"}],"graph_snapshots":[{"event_id":"sha256:b01ec0151837cd10081d706ebb811504d160fe57b9c2a38522e5e9c17799b0a0","target":"graph","created_at":"2026-05-18T04:14:46Z","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 greater than 3. In the paper we mainly determine $\\sum_{k=0}^{[p/4]}\\binom{4k}{2k}(-1)^k$, $\\sum_{k=0}^{[p/3]}\\binom{3k}k, \\sum_{k=0}^{[p/3]}\\binom{3k}k(-1)^k$ and $\\sum_{k=0}^{[p/3]}\\binom{3k}k(-3)^k$ modulo $p$, where $[x]$ is the greatest integer not exceeding $x$.","authors_text":"Zhi-Hong Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-24T13:47:04Z","title":"Congruences involving $\\binom{4k}{2k}$ and $\\binom{3k}k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.4840","kind":"arxiv","version":1},"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":"sha256:73750e2adbd91bc45e88205b30f3ae7574a269c239d8c3bfcd1bbc5697a4c65c","target":"record","created_at":"2026-05-18T04:14:46Z","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":"87f815efdcf17a08d81ee20f7308a1485ffa3912919a7ea0bde793f4eed80632","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-24T13:47:04Z","title_canon_sha256":"8d337bc93e67bb4d23a43411b0307bb6d2f99cc656c7f4076d9fe84c4123df2b"},"schema_version":"1.0","source":{"id":"1108.4840","kind":"arxiv","version":1}},"canonical_sha256":"db5c794335f4eb49667f4d765ca8e4b10cc9af1844b5ac29eab153b29dbcf915","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"db5c794335f4eb49667f4d765ca8e4b10cc9af1844b5ac29eab153b29dbcf915","first_computed_at":"2026-05-18T04:14:46.681823Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:14:46.681823Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"wcgN4Ni/UJxEchp0qYkk5Gf0vqF1HPx1C25GwmhCBztWzMTBKdok0JIe0VikjaCSzkDELqPRlxT7XtxXnMx1BA==","signature_status":"signed_v1","signed_at":"2026-05-18T04:14:46.682260Z","signed_message":"canonical_sha256_bytes"},"source_id":"1108.4840","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:73750e2adbd91bc45e88205b30f3ae7574a269c239d8c3bfcd1bbc5697a4c65c","sha256:b01ec0151837cd10081d706ebb811504d160fe57b9c2a38522e5e9c17799b0a0"],"state_sha256":"d8c2daa019bee8a19819c3eb67e15098ab001fb59d9cf5aa5e95c38efe74cefe"}