{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:6WC5LWJXPVSGPVS7S6HGQ7W73Z","short_pith_number":"pith:6WC5LWJX","schema_version":"1.0","canonical_sha256":"f585d5d9377d6467d65f978e687edfde4d6954095e8baaf7312c34f691196c67","source":{"kind":"arxiv","id":"1810.12102","version":8},"attestation_state":"computed","paper":{"title":"Quadratic residues and quartic residues modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2018-10-22T15:56:13Z","abstract_excerpt":"In this paper we study some products related to quadratic residues and quartic residues modulo primes. 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 $$\\prod^{(p-1)/2}_{i,j=1\\atop p\\nmid 2i^2+5ij+2j^2"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1810.12102","kind":"arxiv","version":8},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-22T15:56:13Z","cross_cats_sorted":[],"title_canon_sha256":"0804c9d7c1771e0f51804ec75acf00ba39c5834de0f75352cd0eaa3a86afab39","abstract_canon_sha256":"0ac65728ccba954f44ee8c3c42da19447d927e9f8da0762d866e71500c80fb3f"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:34:19.411234Z","signature_b64":"1Lefw0ErIUNrz9zfZAJ/EFIbrJLYvL59Dr+ImWEAUlLqbRe0hAPbY/hDaKLaFDN59sTKJr8nbbxKokhr0y1aAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f585d5d9377d6467d65f978e687edfde4d6954095e8baaf7312c34f691196c67","last_reissued_at":"2026-07-05T01:34:19.410790Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:34:19.410790Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quadratic residues and quartic residues modulo primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2018-10-22T15:56:13Z","abstract_excerpt":"In this paper we study some products related to quadratic residues and quartic residues modulo primes. Let $p$ be an odd prime and let $A$ be any integer. We mainly determine completely the product $$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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.12102","kind":"arxiv","version":8},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1810.12102/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1810.12102","created_at":"2026-07-05T01:34:19.410853+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.12102v8","created_at":"2026-07-05T01:34:19.410853+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.12102","created_at":"2026-07-05T01:34:19.410853+00:00"},{"alias_kind":"pith_short_12","alias_value":"6WC5LWJXPVSG","created_at":"2026-07-05T01:34:19.410853+00:00"},{"alias_kind":"pith_short_16","alias_value":"6WC5LWJXPVSGPVS7","created_at":"2026-07-05T01:34:19.410853+00:00"},{"alias_kind":"pith_short_8","alias_value":"6WC5LWJX","created_at":"2026-07-05T01:34:19.410853+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z","json":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z.json","graph_json":"https://pith.science/api/pith-number/6WC5LWJXPVSGPVS7S6HGQ7W73Z/graph.json","events_json":"https://pith.science/api/pith-number/6WC5LWJXPVSGPVS7S6HGQ7W73Z/events.json","paper":"https://pith.science/paper/6WC5LWJX"},"agent_actions":{"view_html":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z","download_json":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z.json","view_paper":"https://pith.science/paper/6WC5LWJX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.12102&json=true","fetch_graph":"https://pith.science/api/pith-number/6WC5LWJXPVSGPVS7S6HGQ7W73Z/graph.json","fetch_events":"https://pith.science/api/pith-number/6WC5LWJXPVSGPVS7S6HGQ7W73Z/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z/action/timestamp_anchor","attest_storage":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z/action/storage_attestation","attest_author":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z/action/author_attestation","sign_citation":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z/action/citation_signature","submit_replication":"https://pith.science/pith/6WC5LWJXPVSGPVS7S6HGQ7W73Z/action/replication_record"}},"created_at":"2026-07-05T01:34:19.410853+00:00","updated_at":"2026-07-05T01:34:19.410853+00:00"}