{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:EKL7WWI5IXE6P24ILMX6CU5USA","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":"5fc80e85c1c8f2c81c68e905de1bb3b36a58147f53f40a15620ec6c19ddc0b34","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-14T19:26:16Z","title_canon_sha256":"37de0ba3f954506de2b7a128f290e1e0e400902a5c7bdba028ec9e25d4cec83c"},"schema_version":"1.0","source":{"id":"1012.3141","kind":"arxiv","version":6}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1012.3141","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"arxiv_version","alias_value":"1012.3141v6","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3141","created_at":"2026-05-18T02:53:15Z"},{"alias_kind":"pith_short_12","alias_value":"EKL7WWI5IXE6","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"EKL7WWI5IXE6P24I","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"EKL7WWI5","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:313ac5ea4746a38c9abf0bd6f0027e9d0c07f74616b81a9c02a89052a9f9ddab","target":"graph","created_at":"2026-05-18T02:53:15Z","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":"In this paper we mainly employ the Zeilberger algorithm to study congruences for sums of terms involving products of three binomial coefficients. Let $p>3$ be a prime. We prove that $$\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^2\\binom{2k}{k+d}}{64^k}\\equiv 0\\pmod{p^2}$$ for all $d\\in\\{0,\\ldots,p-1\\}$ with $d\\equiv (p+1)/2\\pmod2$. If $p\\equiv 1\\pmod4$ and $p=x^2+y^2$ with $x\\equiv 1\\pmod4$ and $y\\equiv 0\\pmod2$, then we show $$\\sum_{k=0}^{p-1}\\frac{\\binom{2k}k^2\\binom{2k}{k+1}}{(-8)^k}\\equiv 2p-2x^2\\pmod{p^2}\\ \\ \\mbox{and}\\ \\ \\sum_{k=0}^{p-1}\\frac{\\binom{2k}k\\binom{2k}{k+1}^2}{(-8)^k}\\equiv-2p\\pmod{p^2}","authors_text":"Zhi-Wei Sun","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-14T19:26:16Z","title":"On sums involving products of three binomial coefficients"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3141","kind":"arxiv","version":6},"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:f5523bf005765484b004f5ac9bc240fe7a5850343ac5a8f6d672f5e923e5d1ed","target":"record","created_at":"2026-05-18T02:53:15Z","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":"5fc80e85c1c8f2c81c68e905de1bb3b36a58147f53f40a15620ec6c19ddc0b34","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-14T19:26:16Z","title_canon_sha256":"37de0ba3f954506de2b7a128f290e1e0e400902a5c7bdba028ec9e25d4cec83c"},"schema_version":"1.0","source":{"id":"1012.3141","kind":"arxiv","version":6}},"canonical_sha256":"2297fb591d45c9e7eb885b2fe153b4901938b4a5cb3e3c9364b797d7f826881b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2297fb591d45c9e7eb885b2fe153b4901938b4a5cb3e3c9364b797d7f826881b","first_computed_at":"2026-05-18T02:53:15.916679Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:53:15.916679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"fhWJ2w/s9n21BWSeq18nQyn+HHJ8+DOyjL1YilTKfgrKP/254sLyfSGQPMCzvnaUA2h1D5YITCT2AmClZCmTAQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:53:15.917397Z","signed_message":"canonical_sha256_bytes"},"source_id":"1012.3141","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f5523bf005765484b004f5ac9bc240fe7a5850343ac5a8f6d672f5e923e5d1ed","sha256:313ac5ea4746a38c9abf0bd6f0027e9d0c07f74616b81a9c02a89052a9f9ddab"],"state_sha256":"31ec047eaf595b552d3ad89d5f5c78813dc008066b1339e40cfe3858d56ee178"}