{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:EWJANM2K4FXHL6M2ORALDJMVXE","short_pith_number":"pith:EWJANM2K","schema_version":"1.0","canonical_sha256":"259206b34ae16e75f99a7440b1a595b92ec985bd451f1f566b179866920aea57","source":{"kind":"arxiv","id":"1011.1902","version":6},"attestation_state":"computed","paper":{"title":"A refinement of a congruence result by van Hamme and Mortenson","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":"2010-11-08T20:55:31Z","abstract_excerpt":"Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\\sum_{k=0}^{(p-1)/2}(4k+1)\\binom{-1/2}k^3\\equiv (-1)^{(p-1)/2}p\\pmod{p^3}.$$ In this paper we show further that \\begin{align*}\\sum_{k=0}^{p-1}(4k+1)\\binom{-1/2}k^3\\equiv &\\sum_{k=0}^{(p-1)/2}(4k+1)\\binom{-1/2}k^3 \\\\\\equiv & (-1)^{(p-1)/2}p+p^3E_{p-3} \\pmod{p^4},\\end{align*}where $E_0,E_1,E_2,\\ldots$ are Euler numbers. We also prove that if $p>3$ then $$\\sum_{k=0}^{(p-1)/2}\\frac{20k+3}{(-2^{10})^k}\\binom{4k}{k,k,k,k}\\equiv(-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2)\\pmod{p^4}.$$"},"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":"1011.1902","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-11-08T20:55:31Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"34ba3bd78cfe8dbe2512db15627a2a6981a4b3ebee156745625f3a3ddad3455e","abstract_canon_sha256":"1569a4a4b7277517a8d5b41f28623eb84005f16844069db89c45ca1ca4541415"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:58:46.852853Z","signature_b64":"+hhKfUXatwOU7vemVx8kS6PxrP7+R5s5RHT3W+S2DpmLZAi7tCfWutkSwtmOtTkLBDjy+u7nPmCVy4C1t9hcBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"259206b34ae16e75f99a7440b1a595b92ec985bd451f1f566b179866920aea57","last_reissued_at":"2026-05-18T02:58:46.851991Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:58:46.851991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A refinement of a congruence result by van Hamme and Mortenson","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":"2010-11-08T20:55:31Z","abstract_excerpt":"Let $p$ be an odd prime. In 2008 E. Mortenson proved van Hamme's following conjecture: $$\\sum_{k=0}^{(p-1)/2}(4k+1)\\binom{-1/2}k^3\\equiv (-1)^{(p-1)/2}p\\pmod{p^3}.$$ In this paper we show further that \\begin{align*}\\sum_{k=0}^{p-1}(4k+1)\\binom{-1/2}k^3\\equiv &\\sum_{k=0}^{(p-1)/2}(4k+1)\\binom{-1/2}k^3 \\\\\\equiv & (-1)^{(p-1)/2}p+p^3E_{p-3} \\pmod{p^4},\\end{align*}where $E_0,E_1,E_2,\\ldots$ are Euler numbers. We also prove that if $p>3$ then $$\\sum_{k=0}^{(p-1)/2}\\frac{20k+3}{(-2^{10})^k}\\binom{4k}{k,k,k,k}\\equiv(-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2)\\pmod{p^4}.$$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1902","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":"1011.1902","created_at":"2026-05-18T02:58:46.852143+00:00"},{"alias_kind":"arxiv_version","alias_value":"1011.1902v6","created_at":"2026-05-18T02:58:46.852143+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1011.1902","created_at":"2026-05-18T02:58:46.852143+00:00"},{"alias_kind":"pith_short_12","alias_value":"EWJANM2K4FXH","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_16","alias_value":"EWJANM2K4FXHL6M2","created_at":"2026-05-18T12:26:06.534383+00:00"},{"alias_kind":"pith_short_8","alias_value":"EWJANM2K","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/EWJANM2K4FXHL6M2ORALDJMVXE","json":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE.json","graph_json":"https://pith.science/api/pith-number/EWJANM2K4FXHL6M2ORALDJMVXE/graph.json","events_json":"https://pith.science/api/pith-number/EWJANM2K4FXHL6M2ORALDJMVXE/events.json","paper":"https://pith.science/paper/EWJANM2K"},"agent_actions":{"view_html":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE","download_json":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE.json","view_paper":"https://pith.science/paper/EWJANM2K","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1011.1902&json=true","fetch_graph":"https://pith.science/api/pith-number/EWJANM2K4FXHL6M2ORALDJMVXE/graph.json","fetch_events":"https://pith.science/api/pith-number/EWJANM2K4FXHL6M2ORALDJMVXE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE/action/storage_attestation","attest_author":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE/action/author_attestation","sign_citation":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE/action/citation_signature","submit_replication":"https://pith.science/pith/EWJANM2K4FXHL6M2ORALDJMVXE/action/replication_record"}},"created_at":"2026-05-18T02:58:46.852143+00:00","updated_at":"2026-05-18T02:58:46.852143+00:00"}