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For any integer $d$ with $p\\nmid d$ and any positive integer $m$, Sun introduced the determinants $$T_m(d,p)=\\det\\left[(i^2+dj^2)^m\\left(\\frac{i^2+dj^2}{p}\\right)\\right]_{1\\leqslant i,j \\leqslant (p-1)/2},$$ and $$D_p^{(m)}= \\det\\left[(i^2-j^2)^m\\left(\\frac{i^2-j^2}{p}\\right)\\right]_{1\\leq i,j\\leq (p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $ \\sqrt{D_p^{(m)}}$ for some $m$. We also confirm some related conjectures posed by Zhi-Wei Sun."},"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":"2407.04556","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-05T14:39:12Z","cross_cats_sorted":[],"title_canon_sha256":"a622545ec4f07cefee8e977796edb60e430acef1fda685eea5c8a7e2d2d45fac","abstract_canon_sha256":"c812935419f26e9809cd928756035e365cfb80391f16b2c74591f11ea51e53ee"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:43:11.728949Z","signature_b64":"3PxTJKmBLqSVugF5pkoXlDtqhWHffhQBNc4CANMHWfoNnJwDxCcBnTTzSM+kEKzqc0zGlKXG3UVwh/6fiqtCDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"66509b4a1e3cfd3527034bff677525c3dbe25ba015782043f96dce3cff1abe3e","last_reissued_at":"2026-07-05T08:43:11.728483Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:43:11.728483Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On certain determinants and the square root of some determinants involving Legendre Symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chen-Kai Ren, Xin-Qi Luo","submitted_at":"2024-07-05T14:39:12Z","abstract_excerpt":"Let $p>3$ be a prime and $(\\frac{.}{p})$ be the Legendre symbol. For any integer $d$ with $p\\nmid d$ and any positive integer $m$, Sun introduced the determinants $$T_m(d,p)=\\det\\left[(i^2+dj^2)^m\\left(\\frac{i^2+dj^2}{p}\\right)\\right]_{1\\leqslant i,j \\leqslant (p-1)/2},$$ and $$D_p^{(m)}= \\det\\left[(i^2-j^2)^m\\left(\\frac{i^2-j^2}{p}\\right)\\right]_{1\\leq i,j\\leq (p-1)/2} .$$ In this paper, we obtain some properties of $T_m (d,p)$ and $ \\sqrt{D_p^{(m)}}$ for some $m$. 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