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Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$f_p(A):=\\prod_{1\\le i,j\\le(p-1)/2\\atop p\\nmid i^2-Aij-j^2}(i^2-Aij-j^2)$$ modulo $p$; for example, if $p\\equiv1\\pmod4$ then $$f_p(A)\\equiv\\begin{cases}-(A^2+4)^{(p-1)/4}\\pmod p&\\text{if}\\ (\\frac{A^2+4}p)=1, \\\\(-A^2-4)^{(p-1)/4}\\pmod p&\\text{if}\\ (\\frac{A^2+4}p)=-1,\\end{cases}$$ where $(\\frac{\\cdot}p)$ denotes the Legendre symbol. We also determine $$\\prod^{(p-1)/2}_{i,j=1\\atop p\\nmid 2i^2+5ij+2j^2","authors_text":"Zhi-Wei Sun","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-22T15:56:13Z","title":"Quadratic residues and quartic residues modulo primes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.12102","kind":"arxiv","version":8},"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":"sha256:60e3d7196267318474dee15a836751173228b1046f5adf637ed6afa4c4a51924","target":"record","created_at":"2026-07-05T01:34:19Z","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":"0ac65728ccba954f44ee8c3c42da19447d927e9f8da0762d866e71500c80fb3f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-22T15:56:13Z","title_canon_sha256":"0804c9d7c1771e0f51804ec75acf00ba39c5834de0f75352cd0eaa3a86afab39"},"schema_version":"1.0","source":{"id":"1810.12102","kind":"arxiv","version":8}},"canonical_sha256":"f585d5d9377d6467d65f978e687edfde4d6954095e8baaf7312c34f691196c67","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f585d5d9377d6467d65f978e687edfde4d6954095e8baaf7312c34f691196c67","first_computed_at":"2026-07-05T01:34:19.410790Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T01:34:19.410790Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"1Lefw0ErIUNrz9zfZAJ/EFIbrJLYvL59Dr+ImWEAUlLqbRe0hAPbY/hDaKLaFDN59sTKJr8nbbxKokhr0y1aAg==","signature_status":"signed_v1","signed_at":"2026-07-05T01:34:19.411234Z","signed_message":"canonical_sha256_bytes"},"source_id":"1810.12102","source_kind":"arxiv","source_version":8}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:60e3d7196267318474dee15a836751173228b1046f5adf637ed6afa4c4a51924","sha256:9ef471b0dfd71af034f45688e107b702d82303a9fca8de3eb43b42b9477e2033"],"state_sha256":"88defeb633e9974aea8294c9e17a63263e499ac7bcb45dba88d11af4ef2e88b5"}