{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2010:4JBBYTVJ3DBRKTAKRWJCGVHOIY","short_pith_number":"pith:4JBBYTVJ","schema_version":"1.0","canonical_sha256":"e2421c4ea9d8c3154c0a8d922354ee46275b5aa8d8d3e9512a836215036dfc6a","source":{"kind":"arxiv","id":"1012.3919","version":1},"attestation_state":"computed","paper":{"title":"Constructing $x^2$ for primes $p=ax^2+by^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2010-12-17T16:38:00Z","abstract_excerpt":"Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=ax^2+by^2$ for some integers $x$ and $y$. Let $\\lambda(a,b;n)$ be given by $q\\prod_{k=1}^\\infty (1-q^{ak})^3(1-q^{bk})^3 = \\sum_{n=1}^\\infty \\lambda(a,b;n)q^n$. In the paper, using Jacobi's identity $\\prod_{n=1}^\\infty (1-q^n)^3 = \\sum_{k=0}^\\infty (-1)^k(2k+1)q^{\\frac{k(k+1)}2}$ we construct $x^2$ in terms of $\\lambda(a,b;n)$. For example, if $2\\nmid ab$ and $p\\nmid ab(ab+1)$, then $(-1)^{\\frac{a+b}2x+\\frac{b+1}2}(4ax^2-2p) = \\lambda(a,b;((ab+1)p-a-b)/8+1)$. We also give formulas for $\\lambda(1,3;n+1),\\lambda(1,7;2n"},"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":"1012.3919","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-12-17T16:38:00Z","cross_cats_sorted":[],"title_canon_sha256":"5b73b01a24a0c747f830898e00cbaa75faae74a831b98776094b64cede209442","abstract_canon_sha256":"4e5cd2ef6e957b14af0d3c59080515de5e604820d992ae52192d26e077cfa044"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:33:03.640862Z","signature_b64":"PBOOIU0D/lBIwOOxOqAY0Jv0qKlcFnMUzL9D8AnVYa7yrOhLRdlEfJ/EeBRf4n+B3WmDqe6TTbX5L+wJ3USMDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e2421c4ea9d8c3154c0a8d922354ee46275b5aa8d8d3e9512a836215036dfc6a","last_reissued_at":"2026-05-18T04:33:03.640243Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:33:03.640243Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Constructing $x^2$ for primes $p=ax^2+by^2$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Hong Sun","submitted_at":"2010-12-17T16:38:00Z","abstract_excerpt":"Let $a$ and $b$ be positive integers and let $p$ be an odd prime such that $p=ax^2+by^2$ for some integers $x$ and $y$. Let $\\lambda(a,b;n)$ be given by $q\\prod_{k=1}^\\infty (1-q^{ak})^3(1-q^{bk})^3 = \\sum_{n=1}^\\infty \\lambda(a,b;n)q^n$. In the paper, using Jacobi's identity $\\prod_{n=1}^\\infty (1-q^n)^3 = \\sum_{k=0}^\\infty (-1)^k(2k+1)q^{\\frac{k(k+1)}2}$ we construct $x^2$ in terms of $\\lambda(a,b;n)$. For example, if $2\\nmid ab$ and $p\\nmid ab(ab+1)$, then $(-1)^{\\frac{a+b}2x+\\frac{b+1}2}(4ax^2-2p) = \\lambda(a,b;((ab+1)p-a-b)/8+1)$. We also give formulas for $\\lambda(1,3;n+1),\\lambda(1,7;2n"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.3919","kind":"arxiv","version":1},"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":"1012.3919","created_at":"2026-05-18T04:33:03.640347+00:00"},{"alias_kind":"arxiv_version","alias_value":"1012.3919v1","created_at":"2026-05-18T04:33:03.640347+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1012.3919","created_at":"2026-05-18T04:33:03.640347+00:00"},{"alias_kind":"pith_short_12","alias_value":"4JBBYTVJ3DBR","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_16","alias_value":"4JBBYTVJ3DBRKTAK","created_at":"2026-05-18T12:26:03.138858+00:00"},{"alias_kind":"pith_short_8","alias_value":"4JBBYTVJ","created_at":"2026-05-18T12:26:03.138858+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/4JBBYTVJ3DBRKTAKRWJCGVHOIY","json":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY.json","graph_json":"https://pith.science/api/pith-number/4JBBYTVJ3DBRKTAKRWJCGVHOIY/graph.json","events_json":"https://pith.science/api/pith-number/4JBBYTVJ3DBRKTAKRWJCGVHOIY/events.json","paper":"https://pith.science/paper/4JBBYTVJ"},"agent_actions":{"view_html":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY","download_json":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY.json","view_paper":"https://pith.science/paper/4JBBYTVJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1012.3919&json=true","fetch_graph":"https://pith.science/api/pith-number/4JBBYTVJ3DBRKTAKRWJCGVHOIY/graph.json","fetch_events":"https://pith.science/api/pith-number/4JBBYTVJ3DBRKTAKRWJCGVHOIY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY/action/storage_attestation","attest_author":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY/action/author_attestation","sign_citation":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY/action/citation_signature","submit_replication":"https://pith.science/pith/4JBBYTVJ3DBRKTAKRWJCGVHOIY/action/replication_record"}},"created_at":"2026-05-18T04:33:03.640347+00:00","updated_at":"2026-05-18T04:33:03.640347+00:00"}