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For example, we prove that \\begin{align*} \\sum_{k=0}^{\\frac{p-1}{2}} {2k\\brack k}_{q^2}^3 \\frac{q^{2k}}{(-q^2;q^2)_k^2 (-q;q)_{2k}^2} &\\equiv 0\\pmod{[p]^2} \\quad\\text{for}\\quad p\\equiv 3\\pmod 4, \\\\ \\sum_{k=0}^{\\frac{p-1}{2}}{2k\\brack k}_{q^3}\\frac{(q;q^3)_k (q^{2};q^3)_{k} q^{3k} }{ (q^{6};q^{6})_k^"},"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":"1408.0512","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-08-03T17:03:06Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"20f094fec99e2e4cf1ab30356e87c527b983c562c30f1960908865cc143a2b4f","abstract_canon_sha256":"f5c6de2e49af07a3303eb728eddda04b7f49311411639feb430a178deaae3693"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:56.883929Z","signature_b64":"EjK5L3enZ67CA5diJtk7OUfrveSRVZpjALQzanFNnf6452Ek8b0zQf9ZK8eqSZQOgmG1v3psweVdvlLgLjY2Bg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4fd9b81ecc689c703f128c5463e62dbc762a54f88e1b362121b99e9065fc4c49","last_reissued_at":"2026-05-18T02:45:56.883440Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:56.883440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Some q-analogues of (super)congruences of Beukers, Van Hamme and 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. 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