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We determine $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+d}$ mod $p^2$ for $d=0,...,p^a$ and $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+\\delta}$ mod $p^3$ for $\\delta=0,1$. We also show that $$C_n^{-1}\\sum_{k=0}^{p^a-1}C_{p^an+k}=1-3(n+1)((p^a-1)/3) (mod p^2)$$ for every n=0,1,2,..., where $C_m$ is the Catalan number $\\binom{2m}{m}/(m+1)$, and (-) is the Legendre symbol."},"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":"0709.1665","kind":"arxiv","version":10},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2007-09-11T17:22:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"ff2bb0c810b3848adef5153a3c4aa55ea992104c8b36d696fd1d2ed6982be124","abstract_canon_sha256":"570a6c102b86401626f2e6ddd28d83c8fc21de80dac603642dcb4142f646c822"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:20:51.469444Z","signature_b64":"QADmL/a3rteD0YB4EHFsYGUk0MvwukInw4Ge5g6JOww/MRSHfLBZA8PM94BUpLuOjGX9DQBX0n3l84FNhl/VCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7bb995ee6e96f867d080f22f3bb802b0cb8f5cc5b56b7a1b83cb9e466231c26e","last_reissued_at":"2026-05-18T04:20:51.468736Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:20:51.468736Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some new congruences for binomial coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Roberto Tauraso, Zhi-Wei Sun","submitted_at":"2007-09-11T17:22:26Z","abstract_excerpt":"In this paper we establish some new congruences involving central binomial coefficients as well as Catalan numbers. Let $p$ be a prime and let $a$ be any positive integer. We determine $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+d}$ mod $p^2$ for $d=0,...,p^a$ and $\\sum_{k=0}^{p^a-1}\\binom{2k}{k+\\delta}$ mod $p^3$ for $\\delta=0,1$. We also show that $$C_n^{-1}\\sum_{k=0}^{p^a-1}C_{p^an+k}=1-3(n+1)((p^a-1)/3) (mod p^2)$$ for every n=0,1,2,..., where $C_m$ is the Catalan number $\\binom{2m}{m}/(m+1)$, and (-) is the Legendre symbol."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0709.1665","kind":"arxiv","version":10},"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":"0709.1665","created_at":"2026-05-18T04:20:51.468850+00:00"},{"alias_kind":"arxiv_version","alias_value":"0709.1665v10","created_at":"2026-05-18T04:20:51.468850+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.0709.1665","created_at":"2026-05-18T04:20:51.468850+00:00"},{"alias_kind":"pith_short_12","alias_value":"PO4ZL3TOS34G","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_16","alias_value":"PO4ZL3TOS34GPUEA","created_at":"2026-05-18T12:25:55.427421+00:00"},{"alias_kind":"pith_short_8","alias_value":"PO4ZL3TO","created_at":"2026-05-18T12:25:55.427421+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/PO4ZL3TOS34GPUEA6IXTXOACWD","json":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD.json","graph_json":"https://pith.science/api/pith-number/PO4ZL3TOS34GPUEA6IXTXOACWD/graph.json","events_json":"https://pith.science/api/pith-number/PO4ZL3TOS34GPUEA6IXTXOACWD/events.json","paper":"https://pith.science/paper/PO4ZL3TO"},"agent_actions":{"view_html":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD","download_json":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD.json","view_paper":"https://pith.science/paper/PO4ZL3TO","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=0709.1665&json=true","fetch_graph":"https://pith.science/api/pith-number/PO4ZL3TOS34GPUEA6IXTXOACWD/graph.json","fetch_events":"https://pith.science/api/pith-number/PO4ZL3TOS34GPUEA6IXTXOACWD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD/action/storage_attestation","attest_author":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD/action/author_attestation","sign_citation":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD/action/citation_signature","submit_replication":"https://pith.science/pith/PO4ZL3TOS34GPUEA6IXTXOACWD/action/replication_record"}},"created_at":"2026-05-18T04:20:51.468850+00:00","updated_at":"2026-05-18T04:20:51.468850+00:00"}