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We also prove three congruences modulo $p^3$ conjectured by Sun, one of which is $$\\sum_{k=0}^{p-1}\\binom{p-1}k\\binom{2k}k((-1)^k-(-3)^{-k})\\equiv \\left(\\frac p3\\right)(3^{p-1}-1)\\ \\pmod{p^3}.$$ In addition, we get some new combinatorial identities."},"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":"1006.3069","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-06-15T19:59:20Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"598ab06044b1a8f861c9aa9629111336d10864cdf6f3cac80f973c39d4b8dc0d","abstract_canon_sha256":"d11b3e4206f389940898f4f4689e4774d5590c0c66d3bd3d51ccae03c51d7edc"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:17:52.677174Z","signature_b64":"0nweUV0V9Tf2Bls8QQbt0F7ITpSScmz6vJoqo3NgkQvh8rX+KDHNtUu40EQRrcciZyAgeld2m1LsagxUEjGPBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c14bcca65b0f9de0a131c243953bfeb5fc5c28f99f56ecbbe8a60edf9130b308","last_reissued_at":"2026-05-18T02:17:52.676449Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:17:52.676449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some congruences involving binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Hui-Qin Cao, Zhi-Wei Sun","submitted_at":"2010-06-15T19:59:20Z","abstract_excerpt":"Binomial coefficients and central trinomial coefficients play important roles in combinatorics. Let $p>3$ be a prime. We show that $$T_{p-1}\\equiv\\left(\\frac p3\\right)3^{p-1}\\ \\pmod{p^2},$$ where the central trinomial coefficient $T_n$ is the constant term in the expansion of $(1+x+x^{-1})^n$. We also prove three congruences modulo $p^3$ conjectured by Sun, one of which is $$\\sum_{k=0}^{p-1}\\binom{p-1}k\\binom{2k}k((-1)^k-(-3)^{-k})\\equiv \\left(\\frac p3\\right)(3^{p-1}-1)\\ \\pmod{p^3}.$$ In addition, we get some new combinatorial identities."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3069","kind":"arxiv","version":4},"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":"1006.3069","created_at":"2026-05-18T02:17:52.676571+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.3069v4","created_at":"2026-05-18T02:17:52.676571+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.3069","created_at":"2026-05-18T02:17:52.676571+00:00"},{"alias_kind":"pith_short_12","alias_value":"YFF4ZJS3B6O6","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_16","alias_value":"YFF4ZJS3B6O6BIJR","created_at":"2026-05-18T12:26:17.028572+00:00"},{"alias_kind":"pith_short_8","alias_value":"YFF4ZJS3","created_at":"2026-05-18T12:26:17.028572+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/YFF4ZJS3B6O6BIJRYJBZKO76WX","json":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX.json","graph_json":"https://pith.science/api/pith-number/YFF4ZJS3B6O6BIJRYJBZKO76WX/graph.json","events_json":"https://pith.science/api/pith-number/YFF4ZJS3B6O6BIJRYJBZKO76WX/events.json","paper":"https://pith.science/paper/YFF4ZJS3"},"agent_actions":{"view_html":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX","download_json":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX.json","view_paper":"https://pith.science/paper/YFF4ZJS3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.3069&json=true","fetch_graph":"https://pith.science/api/pith-number/YFF4ZJS3B6O6BIJRYJBZKO76WX/graph.json","fetch_events":"https://pith.science/api/pith-number/YFF4ZJS3B6O6BIJRYJBZKO76WX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX/action/storage_attestation","attest_author":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX/action/author_attestation","sign_citation":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX/action/citation_signature","submit_replication":"https://pith.science/pith/YFF4ZJS3B6O6BIJRYJBZKO76WX/action/replication_record"}},"created_at":"2026-05-18T02:17:52.676571+00:00","updated_at":"2026-05-18T02:17:52.676571+00:00"}