{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:EKL7WWI5IXE6P24ILMX6CU5USA","short_pith_number":"pith:EKL7WWI5","schema_version":"1.0","canonical_sha256":"2297fb591d45c9e7eb885b2fe153b4901938b4a5cb3e3c9364b797d7f826881b","source":{"kind":"arxiv","id":"1012.3141","version":6},"attestation_state":"computed","paper":{"title":"On sums involving products of three binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2010-12-14T19:26:16Z","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}"},"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":"1012.3141","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-14T19:26:16Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"37de0ba3f954506de2b7a128f290e1e0e400902a5c7bdba028ec9e25d4cec83c","abstract_canon_sha256":"5fc80e85c1c8f2c81c68e905de1bb3b36a58147f53f40a15620ec6c19ddc0b34"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:53:15.917397Z","signature_b64":"fhWJ2w/s9n21BWSeq18nQyn+HHJ8+DOyjL1YilTKfgrKP/254sLyfSGQPMCzvnaUA2h1D5YITCT2AmClZCmTAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2297fb591d45c9e7eb885b2fe153b4901938b4a5cb3e3c9364b797d7f826881b","last_reissued_at":"2026-05-18T02:53:15.916679Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:53:15.916679Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On sums involving products of three binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2010-12-14T19:26:16Z","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}"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3141","kind":"arxiv","version":6},"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":"1012.3141","created_at":"2026-05-18T02:53:15.916807+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.3141v6","created_at":"2026-05-18T02:53:15.916807+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3141","created_at":"2026-05-18T02:53:15.916807+00:00"},{"alias_kind":"pith_short_12","alias_value":"EKL7WWI5IXE6","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_16","alias_value":"EKL7WWI5IXE6P24I","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_8","alias_value":"EKL7WWI5","created_at":"2026-05-18T12:26:06.534383+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/EKL7WWI5IXE6P24ILMX6CU5USA","json":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA.json","graph_json":"https://pith.science/api/pith-number/EKL7WWI5IXE6P24ILMX6CU5USA/graph.json","events_json":"https://pith.science/api/pith-number/EKL7WWI5IXE6P24ILMX6CU5USA/events.json","paper":"https://pith.science/paper/EKL7WWI5"},"agent_actions":{"view_html":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA","download_json":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA.json","view_paper":"https://pith.science/paper/EKL7WWI5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.3141&json=true","fetch_graph":"https://pith.science/api/pith-number/EKL7WWI5IXE6P24ILMX6CU5USA/graph.json","fetch_events":"https://pith.science/api/pith-number/EKL7WWI5IXE6P24ILMX6CU5USA/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA/action/storage_attestation","attest_author":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA/action/author_attestation","sign_citation":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA/action/citation_signature","submit_replication":"https://pith.science/pith/EKL7WWI5IXE6P24ILMX6CU5USA/action/replication_record"}},"created_at":"2026-05-18T02:53:15.916807+00:00","updated_at":"2026-05-18T02:53:15.916807+00:00"}