{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2011:FFRIMMYKOOVYRTW2L3V2MMMHGM","short_pith_number":"pith:FFRIMMYK","schema_version":"1.0","canonical_sha256":"296286330a73ab88ceda5eeba6318733244f5d9edb3d0ea4c297e5a22570526a","source":{"kind":"arxiv","id":"1111.4988","version":4},"attestation_state":"computed","paper":{"title":"p-adic congruences motivated by series","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":"2011-11-21T19:58:22Z","abstract_excerpt":"Let $p>5$ be a prime. 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},$$ where $B_0,B_1,\\ldots$ are Bernoulli number"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1111.4988","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-11-21T19:58:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a94d866e35f8ceb9da7b3a942ca2b4a26f204b0e38d4da92ab3c1ebeb20c1b83","abstract_canon_sha256":"2c5fbcbb0e1f0e4b84812c11657850389d80a70f433c6bfcc40be7af2f086c65"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:05:44.985917Z","signature_b64":"aaZBNnBnk8D/rwm1StK3mMihcIPURrzkRMCR9GP+272J5z2OzOYMHxoRiSLqQ0gvpY0s9lAJwbPGNV+MD1fWDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"296286330a73ab88ceda5eeba6318733244f5d9edb3d0ea4c297e5a22570526a","last_reissued_at":"2026-05-18T03:05:44.985410Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:05:44.985410Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"p-adic congruences motivated by series","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2011-11-21T19:58:22Z","abstract_excerpt":"Let $p>5$ be a prime. Motivated by the known formulae $\\sum_{k=1}^\\infty(-1)^k/(k^3\\binom{2k}{k})=-2\\zeta(3)/5$ and $\\sum_{k=0}^\\infty \\binom{2k}{k}^2/((2k+1)16^k)=4G/\\pi$$ (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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.4988","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":"1111.4988","created_at":"2026-05-18T03:05:44.985492+00:00"},{"alias_kind":"arxiv_version","alias_value":"1111.4988v4","created_at":"2026-05-18T03:05:44.985492+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.4988","created_at":"2026-05-18T03:05:44.985492+00:00"},{"alias_kind":"pith_short_12","alias_value":"FFRIMMYKOOVY","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_16","alias_value":"FFRIMMYKOOVYRTW2","created_at":"2026-05-18T12:26:28.662955+00:00"},{"alias_kind":"pith_short_8","alias_value":"FFRIMMYK","created_at":"2026-05-18T12:26:28.662955+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/FFRIMMYKOOVYRTW2L3V2MMMHGM","json":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM.json","graph_json":"https://pith.science/api/pith-number/FFRIMMYKOOVYRTW2L3V2MMMHGM/graph.json","events_json":"https://pith.science/api/pith-number/FFRIMMYKOOVYRTW2L3V2MMMHGM/events.json","paper":"https://pith.science/paper/FFRIMMYK"},"agent_actions":{"view_html":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM","download_json":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM.json","view_paper":"https://pith.science/paper/FFRIMMYK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1111.4988&json=true","fetch_graph":"https://pith.science/api/pith-number/FFRIMMYKOOVYRTW2L3V2MMMHGM/graph.json","fetch_events":"https://pith.science/api/pith-number/FFRIMMYKOOVYRTW2L3V2MMMHGM/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM/action/storage_attestation","attest_author":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM/action/author_attestation","sign_citation":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM/action/citation_signature","submit_replication":"https://pith.science/pith/FFRIMMYKOOVYRTW2L3V2MMMHGM/action/replication_record"}},"created_at":"2026-05-18T03:05:44.985492+00:00","updated_at":"2026-05-18T03:05:44.985492+00:00"}