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In the paper, by using the quartic reciprocity law we determine $q^{[p/8]}\\mod p$ in terms of $c,d,x$ and $y$, where $[\\cdot]$ is the greatest integer function. We also determine $\\big(\\frac{b+\\sqrt{b^2+4^{\\alpha}}}2\\big)^{\\frac{p-1}4}\\mod p$ for odd $b$ and $(2a+\\sqrt{4a^2+1})^{\\f{p-1}4}\\mod p$ for $a\\in\\Bb"},"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":"1108.3027","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-08-15T16:14:02Z","cross_cats_sorted":[],"title_canon_sha256":"fba4420431d266264e3c9ce3315997884a3776f65e5d7e38b6973f17c78d761d","abstract_canon_sha256":"da4ff74af1915b49db4df98963b26294a49204d76f4babc81917d3b10efbfea8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:45:10.816716Z","signature_b64":"r5OEKI7rzcQarNDzj4XU/MB0Ql9Uhlkym+urpFxR9fVewXDqYLq0mzC99Sq/v7N0MLnXiOuSCo4kbiy8PJezCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea27271930488c91e523c089c4ee6563b90051df817d9751f478ff7e4510231a","last_reissued_at":"2026-05-18T03:45:10.816118Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:45:10.816118Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quartic, octic residues and binary quadratic forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2011-08-15T16:14:02Z","abstract_excerpt":"Let $\\Bbb Z$ be the set of integers, and let $(m,n)$ be the greatest common divisor of integers $m$ and $n$. Let $p\\equiv 1\\mod 4$ be a prime, $q\\in\\Bbb Z$, $2\\nmid q$ and $p=c^2+d^2=x^2+qy^2$ with $c,d,x,y\\in\\Bbb Z$ and $c\\e 1\\mod 4$. Suppose that $(c,x+d)=1$ or $(d,x+c)$ is a power of 2. In the paper, by using the quartic reciprocity law we determine $q^{[p/8]}\\mod p$ in terms of $c,d,x$ and $y$, where $[\\cdot]$ is the greatest integer function. We also determine $\\big(\\frac{b+\\sqrt{b^2+4^{\\alpha}}}2\\big)^{\\frac{p-1}4}\\mod p$ for odd $b$ and $(2a+\\sqrt{4a^2+1})^{\\f{p-1}4}\\mod p$ for $a\\in\\Bb"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1108.3027","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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":"1108.3027","created_at":"2026-05-18T03:45:10.816202+00:00"},{"alias_kind":"arxiv_version","alias_value":"1108.3027v4","created_at":"2026-05-18T03:45:10.816202+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1108.3027","created_at":"2026-05-18T03:45:10.816202+00:00"},{"alias_kind":"pith_short_12","alias_value":"5ITSOGJQJCGJ","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_16","alias_value":"5ITSOGJQJCGJDZJD","created_at":"2026-05-18T12:26:20.644004+00:00"},{"alias_kind":"pith_short_8","alias_value":"5ITSOGJQ","created_at":"2026-05-18T12:26:20.644004+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/5ITSOGJQJCGJDZJDYCE4J3TFMO","json":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO.json","graph_json":"https://pith.science/api/pith-number/5ITSOGJQJCGJDZJDYCE4J3TFMO/graph.json","events_json":"https://pith.science/api/pith-number/5ITSOGJQJCGJDZJDYCE4J3TFMO/events.json","paper":"https://pith.science/paper/5ITSOGJQ"},"agent_actions":{"view_html":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO","download_json":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO.json","view_paper":"https://pith.science/paper/5ITSOGJQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1108.3027&json=true","fetch_graph":"https://pith.science/api/pith-number/5ITSOGJQJCGJDZJDYCE4J3TFMO/graph.json","fetch_events":"https://pith.science/api/pith-number/5ITSOGJQJCGJDZJDYCE4J3TFMO/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO/action/storage_attestation","attest_author":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO/action/author_attestation","sign_citation":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO/action/citation_signature","submit_replication":"https://pith.science/pith/5ITSOGJQJCGJDZJDYCE4J3TFMO/action/replication_record"}},"created_at":"2026-05-18T03:45:10.816202+00:00","updated_at":"2026-05-18T03:45:10.816202+00:00"}