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In this paper we mainly evaluate $$T_p^{(\\delta)}(a,b,x):=\\det\\left[x+\\tan\\pi\\frac{aj^2+bk^2}p\\right]_{\\delta\\le j,k\\le (p-1)/2}\\ \\ (\\delta=0,1).$$ For example, in the case $p\\equiv3\\pmod4$ we show that $T_p^{(1)}(a,b,0)=0$ and $$T_p^{(0)}(a,b,x)=\\begin{cases} 2^{(p-1)/2}p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=1, \\\\p^{(p+1)/4}&\\text{if}\\ (\\frac{ab}p)=-1,\\end{cases}$$ where $(\\frac{\\cdot}p)$ is the Legendre symbol. When $(\\frac{-ab}p)=-1$, we also evaluate the determinant $\\det[x+\\cot\\pi\\frac{aj^2+bk^2}p]_{1\\le j,k\\le(p-1)/2}.$ In ","authors_text":"Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-15T15:59:29Z","title":"On some determinants involving the tangent function"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1901.04837","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:d90c03af70bc318d81b0bb12b0ec1771e9b42fcce3395efb6b4ddcd80aa0971c","target":"record","created_at":"2026-07-05T07:27:01Z","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":"e671c1a94d67063d27020fe300576c467cef6bb6fb1b463550b1069a745ac77a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2019-01-15T15:59:29Z","title_canon_sha256":"fce341c4e00ac73e8b16e763d6392a2ae7bc7e8bcaaa4b53ade898c0fe944aba"},"schema_version":"1.0","source":{"id":"1901.04837","kind":"arxiv","version":6}},"canonical_sha256":"6377d7c5e5b6be0ea339a45308c5305b8a015eb4274a2ae77be98cea1e26e748","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6377d7c5e5b6be0ea339a45308c5305b8a015eb4274a2ae77be98cea1e26e748","first_computed_at":"2026-07-05T07:27:01.069827Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T07:27:01.069827Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"A7N9f8BLTP1yEtcAZdmWVuDnWPMlwI7HHoF5SpLxAAGAcTrX8yv+2BktvkhqGHkHrpGytCxO7h2QggtZthAXCA==","signature_status":"signed_v1","signed_at":"2026-07-05T07:27:01.070308Z","signed_message":"canonical_sha256_bytes"},"source_id":"1901.04837","source_kind":"arxiv","source_version":6}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d90c03af70bc318d81b0bb12b0ec1771e9b42fcce3395efb6b4ddcd80aa0971c","sha256:326130c33eb6d09353e0da52a0b35ab4f305a860e671470ed46c98f1986e8749"],"state_sha256":"bf571b69e2a850cd984243b5d1ca95e8794c7c2949f9dae77a6bc8ef0476f937"}