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Motivated by the known formulae $\\sum_{k=1}^\\infty(-1)^k/(k^3\\binom{2k}{k})=-2\\zeta(3)/5$ and $\\sum_{k=0}^\\infty \\binom{2k}{k}^2/((2k+1)16^k)=4G/\\pi$$ (where $G=\\sum_{k=0}^\\infty(-1)^k/(2k+1)^2$ is the Catalan constant), we show that $$\\sum_{k=1}^{(p-1)/2}\\frac{(-1)^k}{k^3\\binom{2k}{k}}\\equiv-2B_{p-3}\\pmod{p},$$ $$\\sum_{k=(p+1)/2}^{p-1}\\frac{\\binom{2k}{k}^2}{(2k+1)16^k}\\equiv-\\frac 7{4}p^2B_{p-3}\\pmod{p^3}$$, and $$\\sum_{k=0}^{(p-3)/2}\\frac{\\binom{2k}{k}^2}{(2k+1)16^k} \\equiv-2q_p(2)-pq_p(2)^2+\\frac{5}{12}p^2B_{p-3}\\pmod{p^3},$$ where $B_0,B_1,\\ldots$ are Bernoulli number","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":"2011-11-21T19:58:22Z","title":"p-adic congruences motivated by series"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4988","kind":"arxiv","version":4},"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:c6e6a2144267c74947196f59a675b11d9e9dc53686d0b6b773409c63df3388d8","target":"record","created_at":"2026-05-18T03:05:44Z","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":"2c5fbcbb0e1f0e4b84812c11657850389d80a70f433c6bfcc40be7af2f086c65","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-21T19:58:22Z","title_canon_sha256":"a94d866e35f8ceb9da7b3a942ca2b4a26f204b0e38d4da92ab3c1ebeb20c1b83"},"schema_version":"1.0","source":{"id":"1111.4988","kind":"arxiv","version":4}},"canonical_sha256":"296286330a73ab88ceda5eeba6318733244f5d9edb3d0ea4c297e5a22570526a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"296286330a73ab88ceda5eeba6318733244f5d9edb3d0ea4c297e5a22570526a","first_computed_at":"2026-05-18T03:05:44.985410Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:05:44.985410Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"aaZBNnBnk8D/rwm1StK3mMihcIPURrzkRMCR9GP+272J5z2OzOYMHxoRiSLqQ0gvpY0s9lAJwbPGNV+MD1fWDw==","signature_status":"signed_v1","signed_at":"2026-05-18T03:05:44.985917Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.4988","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c6e6a2144267c74947196f59a675b11d9e9dc53686d0b6b773409c63df3388d8","sha256:190db660ccb6997708d2412863086fd765db4fdd786882c3bd650b7362505c99"],"state_sha256":"cefecb4f5532203d22887e228f03661179425e7c179ba9eb6535b3ffd0a737b8"}