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For example, for any prime $p>3$, we show that $$\\sum_{k=0}^{p-1}\\frac{\\binom{4k}{2k+1}\\binom{2k}k}{48^k}\\equiv0\\pmod{p^2},$$ and $$ \\sum_{k=0}^{p-1}\\frac{\\binom{2k}{k}\\binom{3k}{k}}{24^k}\\equiv\\begin{cases}\\binom{(2p-2)/3}{(p-1)/3}\\pmod{p^2}\\ &\\mbox{if}\\ p\\equiv1\\pmod{3},\\\\ p/\\binom{(2p+2)/3}{(p+1)/3}\\pmod{p^2}\\ &\\mbox{if}\\ p\\equiv2\\pmod{3}.\\end{cases} $$ We also obtain some other results of such types."},"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":"2006.02918","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2020-06-04T15:01:22Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"7ae5d1f53d3f511939e36f5153b5c51c83c2bd154a95f0eae922ec396dd42eaf","abstract_canon_sha256":"f396c122106de67448058d38b8ee3b377a0acadb4ae4253f2cbd863bedb3780f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:04:01.257653Z","signature_b64":"G0Kyu8+ACJFXU+6IhHtFbRWOvOOdIhiMG5yNtDn9Brtp+nqm0uWEYTiL/EudIa4wS5ZIax4vCua0MwfUYD+SCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ba0c257f19e7cb5ee8511c821a56a9cef8aab6589afd1f677a8f1bc2a699205","last_reissued_at":"2026-07-05T03:04:01.257174Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:04:01.257174Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Proof of some conjectural hypergeometric supercongruences via curious identities","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Chen Wang, Zhi-Wei Sun","submitted_at":"2020-06-04T15:01:22Z","abstract_excerpt":"In this paper, we prove several supercongruences conjectured by Z.-W. Sun ten years ago via certain strange hypergeometric identities. For example, for any prime $p>3$, we show that $$\\sum_{k=0}^{p-1}\\frac{\\binom{4k}{2k+1}\\binom{2k}k}{48^k}\\equiv0\\pmod{p^2},$$ and $$ \\sum_{k=0}^{p-1}\\frac{\\binom{2k}{k}\\binom{3k}{k}}{24^k}\\equiv\\begin{cases}\\binom{(2p-2)/3}{(p-1)/3}\\pmod{p^2}\\ &\\mbox{if}\\ p\\equiv1\\pmod{3},\\\\ p/\\binom{(2p+2)/3}{(p+1)/3}\\pmod{p^2}\\ &\\mbox{if}\\ p\\equiv2\\pmod{3}.\\end{cases} $$ We also obtain some other results of such types."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.02918","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2006.02918/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2006.02918","created_at":"2026-07-05T03:04:01.257240+00:00"},{"alias_kind":"arxiv_version","alias_value":"2006.02918v3","created_at":"2026-07-05T03:04:01.257240+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2006.02918","created_at":"2026-07-05T03:04:01.257240+00:00"},{"alias_kind":"pith_short_12","alias_value":"NOQMEV7RTZ6L","created_at":"2026-07-05T03:04:01.257240+00:00"},{"alias_kind":"pith_short_16","alias_value":"NOQMEV7RTZ6LL3UF","created_at":"2026-07-05T03:04:01.257240+00:00"},{"alias_kind":"pith_short_8","alias_value":"NOQMEV7R","created_at":"2026-07-05T03:04:01.257240+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/NOQMEV7RTZ6LL3UFCHECDJLKTT","json":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT.json","graph_json":"https://pith.science/api/pith-number/NOQMEV7RTZ6LL3UFCHECDJLKTT/graph.json","events_json":"https://pith.science/api/pith-number/NOQMEV7RTZ6LL3UFCHECDJLKTT/events.json","paper":"https://pith.science/paper/NOQMEV7R"},"agent_actions":{"view_html":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT","download_json":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT.json","view_paper":"https://pith.science/paper/NOQMEV7R","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2006.02918&json=true","fetch_graph":"https://pith.science/api/pith-number/NOQMEV7RTZ6LL3UFCHECDJLKTT/graph.json","fetch_events":"https://pith.science/api/pith-number/NOQMEV7RTZ6LL3UFCHECDJLKTT/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT/action/timestamp_anchor","attest_storage":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT/action/storage_attestation","attest_author":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT/action/author_attestation","sign_citation":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT/action/citation_signature","submit_replication":"https://pith.science/pith/NOQMEV7RTZ6LL3UFCHECDJLKTT/action/replication_record"}},"created_at":"2026-07-05T03:04:01.257240+00:00","updated_at":"2026-07-05T03:04:01.257240+00:00"}