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For example, for any odd prime p, we show that the known congruence \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}^2}{16^k} \\equiv (-1)^{\\frac{p-1}{2}}\\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \\sum_{k=0}^{p-1}\\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\\varepsilon)k} &\\equiv (-1)^{\\frac{p-1}{2}}q^{\\frac{(p^2-1)\\varepsilon}{4}}\\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \\varepsilon=\\pm1. Several related conjectures are also proposed."},"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":"1401.5978","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-01-23T14:06:26Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"a31f186e5fc91393485d9b4fabfedfadbca75cfe2eda7f73cf8cb1989f81d55f","abstract_canon_sha256":"ae975325b57b6021b4fd0be88c50ebd2ce2e35a38ccb8eda29a55579f55d5d3c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:58.047492Z","signature_b64":"YA635uMeiCgm7Fp75tLlLCkJ2gH1pdlKzHnhov1vUWYgYoAGOEj0/cPQ8To1rJmccapcF4fH+sp4EGWaDR2DCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"92c98759d441d1e03907746654adceebb2cb2fcfe4f3f77efdafae8831a33fe7","last_reissued_at":"2026-05-18T02:45:58.047052Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:58.047052Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some q-analogues of supercongruences of Rodriguez-Villegas","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Jiang Zeng, Victor J. W. Guo","submitted_at":"2014-01-23T14:06:26Z","abstract_excerpt":"We study different q-analogues and generalizations of the ex-conjectures of Rodriguez-Villegas. For example, for any odd prime p, we show that the known congruence \\sum_{k=0}^{p-1}\\frac{{2k\\choose k}^2}{16^k} \\equiv (-1)^{\\frac{p-1}{2}}\\pmod{p^2} has the following two nice q-analogues with [p]=1+q+...+q^{p-1}: \\sum_{k=0}^{p-1}\\frac{(q;q^2)_k^2}{(q^2;q^2)_k^2}q^{(1+\\varepsilon)k} &\\equiv (-1)^{\\frac{p-1}{2}}q^{\\frac{(p^2-1)\\varepsilon}{4}}\\pmod{[p]^2}, where (a;q)_0=1, (a;q)_n=(1-a)(1-aq)...(1-aq^{n-1}) for n=1,2,..., and \\varepsilon=\\pm1. Several related conjectures are also proposed."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.5978","kind":"arxiv","version":2},"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":"1401.5978","created_at":"2026-05-18T02:45:58.047121+00:00"},{"alias_kind":"arxiv_version","alias_value":"1401.5978v2","created_at":"2026-05-18T02:45:58.047121+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1401.5978","created_at":"2026-05-18T02:45:58.047121+00:00"},{"alias_kind":"pith_short_12","alias_value":"SLEYOWOUIHI6","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_16","alias_value":"SLEYOWOUIHI6AOIH","created_at":"2026-05-18T12:28:49.207871+00:00"},{"alias_kind":"pith_short_8","alias_value":"SLEYOWOU","created_at":"2026-05-18T12:28:49.207871+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/SLEYOWOUIHI6AOIHORTFJLOO5O","json":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O.json","graph_json":"https://pith.science/api/pith-number/SLEYOWOUIHI6AOIHORTFJLOO5O/graph.json","events_json":"https://pith.science/api/pith-number/SLEYOWOUIHI6AOIHORTFJLOO5O/events.json","paper":"https://pith.science/paper/SLEYOWOU"},"agent_actions":{"view_html":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O","download_json":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O.json","view_paper":"https://pith.science/paper/SLEYOWOU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1401.5978&json=true","fetch_graph":"https://pith.science/api/pith-number/SLEYOWOUIHI6AOIHORTFJLOO5O/graph.json","fetch_events":"https://pith.science/api/pith-number/SLEYOWOUIHI6AOIHORTFJLOO5O/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O/action/storage_attestation","attest_author":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O/action/author_attestation","sign_citation":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O/action/citation_signature","submit_replication":"https://pith.science/pith/SLEYOWOUIHI6AOIHORTFJLOO5O/action/replication_record"}},"created_at":"2026-05-18T02:45:58.047121+00:00","updated_at":"2026-05-18T02:45:58.047121+00:00"}