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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 $$\\sum_{k=0}^{(p-1)/2}\\frac{20k+3}{(-2^{10})^k}\\binom{4k}{k,k,k,k}\\equiv(-1)^{(p-1)/2}p(2^{p-1}+2-(2^{p-1}-1)^2)\\pmod{p^4}.$$","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":"2010-11-08T20:55:31Z","title":"A refinement of a congruence result by van Hamme and Mortenson"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1011.1902","kind":"arxiv","version":6},"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":"sha256:377b20721a8c7f894eedc4bab38758afe7c2619ca6a946d2d0d355925402a945","target":"record","created_at":"2026-05-18T02:58:46Z","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":"1569a4a4b7277517a8d5b41f28623eb84005f16844069db89c45ca1ca4541415","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-11-08T20:55:31Z","title_canon_sha256":"34ba3bd78cfe8dbe2512db15627a2a6981a4b3ebee156745625f3a3ddad3455e"},"schema_version":"1.0","source":{"id":"1011.1902","kind":"arxiv","version":6}},"canonical_sha256":"259206b34ae16e75f99a7440b1a595b92ec985bd451f1f566b179866920aea57","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"259206b34ae16e75f99a7440b1a595b92ec985bd451f1f566b179866920aea57","first_computed_at":"2026-05-18T02:58:46.851991Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:58:46.851991Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"+hhKfUXatwOU7vemVx8kS6PxrP7+R5s5RHT3W+S2DpmLZAi7tCfWutkSwtmOtTkLBDjy+u7nPmCVy4C1t9hcBA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:58:46.852853Z","signed_message":"canonical_sha256_bytes"},"source_id":"1011.1902","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:377b20721a8c7f894eedc4bab38758afe7c2619ca6a946d2d0d355925402a945","sha256:605c0310a2a9914bbdc01694122514c0331a0d48328784d376b536e822679d15"],"state_sha256":"190cac8efb99a274113fa94138b0ccc088d472299e7845c9123a74d67f422e61"}