{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2019:OSZ3WZMFPNOAZZE45UXQYZORIB","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"baa7c7730401b01e6fbc3775005546a1d72a2e069abc5f36ea9d502f7b718453","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-09T08:57:30Z","title_canon_sha256":"d2203d827d76d901a501df1857b647e4188050916b53a8f834eb8b56b08251d0"},"schema_version":"1.0","source":{"id":"1903.03766","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1903.03766","created_at":"2026-05-17T23:51:40Z"},{"alias_kind":"arxiv_version","alias_value":"1903.03766v1","created_at":"2026-05-17T23:51:40Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1903.03766","created_at":"2026-05-17T23:51:40Z"},{"alias_kind":"pith_short_12","alias_value":"OSZ3WZMFPNOA","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_16","alias_value":"OSZ3WZMFPNOAZZE4","created_at":"2026-05-18T12:33:24Z"},{"alias_kind":"pith_short_8","alias_value":"OSZ3WZMF","created_at":"2026-05-18T12:33:24Z"}],"graph_snapshots":[{"event_id":"sha256:ce426961ee4984103db815d24e296911b56809393398d8e84de76f892c75595c","target":"graph","created_at":"2026-05-17T23:51:40Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We establish some supercongruences related to a supercongruence of Van Hamme, such as \\begin{align*} \\sum_{k=0}^{(p+1)/2} (-1)^k (4k-1)\\frac{(-\\frac{1}{2})_k^3}{k!^3} &\\equiv p(-1)^{(p+1)/2}+p^3(2-E_{p-3})\\pmod{p^{4}},\\\\ \\sum_{k=0}^{(p+1)/2} (4k-1)^5 \\frac{(-\\frac{1}{2})_k^4}{k!^4} &\\equiv 16p\\pmod{p^{4}}, \\end{align*} where $p$ is an odd prime and $E_{p-3}$ is the $(p-3)$-th Euler number. Our proof uses some congruences of Z.-W. Sun, the Wilf--Zeilberger method, Whipple's $_7F_6$ transformation, and the software package {\\tt Sigma} developed by Schneider. We also put forward two related conje","authors_text":"Ji-Cai Liu, Victor J. W. Guo","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-09T08:57:30Z","title":"Some congruences related to a congruence of Van Hamme"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1903.03766","kind":"arxiv","version":1},"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":"sha256:24f1c569ac9aa86d533ebcf539d353ce826b4f6a95f8e9b53bed108fa1ac04f2","target":"record","created_at":"2026-05-17T23:51:40Z","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":"baa7c7730401b01e6fbc3775005546a1d72a2e069abc5f36ea9d502f7b718453","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2019-03-09T08:57:30Z","title_canon_sha256":"d2203d827d76d901a501df1857b647e4188050916b53a8f834eb8b56b08251d0"},"schema_version":"1.0","source":{"id":"1903.03766","kind":"arxiv","version":1}},"canonical_sha256":"74b3bb65857b5c0ce49ced2f0c65d1405f58f3be8e572490e43c6eb14e279257","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"74b3bb65857b5c0ce49ced2f0c65d1405f58f3be8e572490e43c6eb14e279257","first_computed_at":"2026-05-17T23:51:40.990241Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:51:40.990241Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0sScwPD5SYkVdkVLui8yLJ0r1U82/J3RrP3AiKmQqegEf6z4Bx+1yyUp9E3xHXbvw40Eta/rHCwu7rtlmNKJAg==","signature_status":"signed_v1","signed_at":"2026-05-17T23:51:40.990761Z","signed_message":"canonical_sha256_bytes"},"source_id":"1903.03766","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:24f1c569ac9aa86d533ebcf539d353ce826b4f6a95f8e9b53bed108fa1ac04f2","sha256:ce426961ee4984103db815d24e296911b56809393398d8e84de76f892c75595c"],"state_sha256":"27cbb2aba76b776b853dd87424ba6cca585a378dfbff064bbf2ff1bf5ca618d4"}