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In this paper, we deduce some residue properties for the determinant $S_{m,k}(d,p)$ as a generalization of certain results of Sun. Using these, we further prove some conjectures of Sun related to $$\\left(\\frac{\\sqrt{S_{1+\\frac{p-1}{2},2}(-1,p)}}{p}\\right) \\text{ and } \\left(\\frac{\\sqrt"},"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":"2407.07085","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-09T17:56:57Z","cross_cats_sorted":[],"title_canon_sha256":"c609a826ea83e575fab4944bba517c7c7d15ad4d4f37fb6ee1428303d0745796","abstract_canon_sha256":"d70026765912397504f588b63d882e9ddc587cc21c1f3f1af3f05a1ef3a1c77a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T08:41:56.733210Z","signature_b64":"NqsDlPYhXI2VUYCadr5OLgi6VnuTqnIHOm7NgYDi98pr7woHUaDC0Ui9avO55QiDZCYj8pL5hvmQlmYESsl2Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"005e3f9a9eec81b7e061c31e440d95c606a456a6f571169f33fe5398dff8c499","last_reissued_at":"2026-07-05T08:41:56.732824Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T08:41:56.732824Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On some conjectural determinants of Sun involving residues","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gautam Kalita, Rituparna Chaliha","submitted_at":"2024-07-09T17:56:57Z","abstract_excerpt":"For an odd prime $p$ and integers $d, k, m$ with gcd$(p,d)=1$ and $2\\leq k\\leq \\frac{p-1}{2}$, we consider the determinant\n  \\begin{equation*}\n  S_{m,k}(d,p) = \\left|(\\alpha_i - \\alpha_j)^m\\right|_{1 \\leq i,j \\leq \\frac{p-1}{k}},\n  \\end{equation*}\n  where $\\alpha_i$ are distinct $k$-th power residues modulo $p$. In this paper, we deduce some residue properties for the determinant $S_{m,k}(d,p)$ as a generalization of certain results of Sun. 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