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For the determinants $$S_m(d,p)=\\det\\left[(i^2+dj^2)^{m}\\right]_{1\\leqslant i,j \\leqslant (p-1)/2}\\ \\ \\left(\\frac{p-1}2\\leqslant m\\leqslant p-1\\right),$$ Sun recently determined $S_m(d,p)$ modulo $p$ when $m\\in\\{p-2,p-3\\}$ and $(\\frac {-d}p)=-1$. In this paper, we obtain $S_{p-2}(d,p)$ modulo $p$ in the remaining case $(\\frac{-d}p)=1$, and determine the Legendre symbols $(\\frac{S_{p-3}\\,(d,p)}p)$ and $(\\frac{S_{p-4}\\,(d,p)}p)$ in some special cases.","authors_text":"Chen-Kai Ren, Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-17T16:43:17Z","title":"On some determinants arising from quadratic residues"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2404.11547","kind":"arxiv","version":1},"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:a0def1152ef58a5206bc18c8f164aa9e0a55e3b6dcc7b6092a426be8c47a5eb5","target":"record","created_at":"2026-07-05T08:09:10Z","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":"16c541a50b4841a43c2ec1ecc66ad06c312cb829d92fec47e0179ee3fac5f533","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-04-17T16:43:17Z","title_canon_sha256":"4230feb43512ec1d621a637b29c1b8ce7a8edc2b7aa8502095e8a1b61c1e3f4d"},"schema_version":"1.0","source":{"id":"2404.11547","kind":"arxiv","version":1}},"canonical_sha256":"cf151f2a9dd8a4bffc0f50d1429d29f77c8b4d94a3c3951904e15bc8ff6c5042","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"cf151f2a9dd8a4bffc0f50d1429d29f77c8b4d94a3c3951904e15bc8ff6c5042","first_computed_at":"2026-07-05T08:09:10.317193Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T08:09:10.317193Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"BNZfU5/+w5WLEYx+nLvBfd8YfUDy85bfmdCD+HFpe+K3PxhrG7/KXuOCxOXRLv9sNiM4BnHfHqcNU9bDov+qDQ==","signature_status":"signed_v1","signed_at":"2026-07-05T08:09:10.317732Z","signed_message":"canonical_sha256_bytes"},"source_id":"2404.11547","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a0def1152ef58a5206bc18c8f164aa9e0a55e3b6dcc7b6092a426be8c47a5eb5","sha256:d7cd1553e7a0e781833dae1d6211213b8ede01c35a2189ef7af386739dd73de2"],"state_sha256":"b8f29642d294aa60344b1532e776aa715cd11f1cec8148020b9285c007a3a248"}