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For $b,c\\in\\mathbb Z$, Sun introduced the determinant $$D_p(b,c)=\\left|(i^2+bij+cj^2)^{p-2}\\right|_{1\\leqslant i,j \\leqslant p-1},$$ and investigated the Legendre symbol $(\\frac{D_p(b,c)}p)$. Recently Wu, She and Ni proved that $(\\frac{D_p(1,1)}p)=(\\frac {-2}p)$ if $p\\equiv2\\pmod 3$, which confirms a previous conjecture of Sun. In this paper we determine $(\\frac{D_p(1,1)}p)$ in the case $p\\equiv1\\pmod3$. Sun proved that $D_p(2,2)\\equiv0\\pmod p$ if $p\\equiv3\\pmod4$, in contrast we prove that $(\\frac{D_p(2,2)}p)=1$ if $p\\equiv1\\pmod8$, and $(\\frac{D_p(2,2)}p)=0$ if $p\\eq","authors_text":"Xin-Qi Luo, Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-10-26T14:19:47Z","title":"Legendre symbols related to certain determinants"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2210.14741","kind":"arxiv","version":3},"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":"sha256:794317e229dc26e16dd5b286435e85806c5aa093009bdcf245ea832f403b3336","target":"record","created_at":"2026-07-05T06:11:42Z","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":"1902f60d13791eeeadff01c0139a3a262492c53cba994b184b5aa4c4cb2cc8c6","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-10-26T14:19:47Z","title_canon_sha256":"7f133c8fc260a21de0e46c37dd9d4ccc63ab4644430ba292369ad753d1250e92"},"schema_version":"1.0","source":{"id":"2210.14741","kind":"arxiv","version":3}},"canonical_sha256":"14d26e7b082a21ad7576846823244a282e545a1c2851a1ca8dfbf4a6c1eaae40","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14d26e7b082a21ad7576846823244a282e545a1c2851a1ca8dfbf4a6c1eaae40","first_computed_at":"2026-07-05T06:11:42.940253Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T06:11:42.940253Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yWia4Mfv35uY2zIM778wr6HR13dxabHPHZT5bINUQl9+VC6nQDYJXnwVNrKomIfIGJNnxNq2fNawFvyNts+7AA==","signature_status":"signed_v1","signed_at":"2026-07-05T06:11:42.940729Z","signed_message":"canonical_sha256_bytes"},"source_id":"2210.14741","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:794317e229dc26e16dd5b286435e85806c5aa093009bdcf245ea832f403b3336","sha256:f8d33d87de5a2e9a5f89630633f2f5037194ce330a257e1cc1f363b67a95a9c5"],"state_sha256":"f121a4f0f2f8d34644064f621c48860f30d4d1298e3b587330e0a56e94503d99"}