{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2000:HOKXIVCY3XOPJ5LZPTTMMNE7A4","short_pith_number":"pith:HOKXIVCY","schema_version":"1.0","canonical_sha256":"3b95745458dddcf4f5797ce6c6349f071ea38525970f4dd36e81c61c41c1d6f0","source":{"kind":"arxiv","id":"math/0001190","version":2},"attestation_state":"computed","paper":{"title":"Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Mc Laughlin","submitted_at":"2000-01-31T00:00:00Z","abstract_excerpt":"For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\\{F_{i}:=F_{i}(t,X_{1},..., X_{\\lfloor \\frac{n+1}{2} \\rfloor})\\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where $C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\\lfloor \\frac{n+1}{2} \\rfloor}$) and the fundamental solution can be written "},"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":"math/0001190","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2000-01-31T00:00:00Z","cross_cats_sorted":[],"title_canon_sha256":"88d9b4839ade0888563bacb6cd20d7a25de7d2c58b97143e7554cb479f75ef93","abstract_canon_sha256":"babd4f1000a3f519429f78364120e6e71aa461740905cf14b153c4c3b80d5d5a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:56.641243Z","signature_b64":"7kuufWIjVXNOJZPpThArDmstFAyuTPRzvvTi7ar1VRYWLUS+qGiwLncmgMAMud5UNPx9CHThaLd+nHTf8J7ZCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3b95745458dddcf4f5797ce6c6349f071ea38525970f4dd36e81c61c41c1d6f0","last_reissued_at":"2026-05-17T23:56:56.640634Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:56.640634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"James Mc Laughlin","submitted_at":"2000-01-31T00:00:00Z","abstract_excerpt":"For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\\{F_{i}:=F_{i}(t,X_{1},..., X_{\\lfloor \\frac{n+1}{2} \\rfloor})\\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where $C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\\lfloor \\frac{n+1}{2} \\rfloor}$) and the fundamental solution can be written "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0001190","kind":"arxiv","version":2},"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":"math/0001190","created_at":"2026-05-17T23:56:56.640726+00:00"},{"alias_kind":"arxiv_version","alias_value":"math/0001190v2","created_at":"2026-05-17T23:56:56.640726+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.math/0001190","created_at":"2026-05-17T23:56:56.640726+00:00"},{"alias_kind":"pith_short_12","alias_value":"HOKXIVCY3XOP","created_at":"2026-05-18T12:25:49.631198+00:00"},{"alias_kind":"pith_short_16","alias_value":"HOKXIVCY3XOPJ5LZ","created_at":"2026-05-18T12:25:49.631198+00:00"},{"alias_kind":"pith_short_8","alias_value":"HOKXIVCY","created_at":"2026-05-18T12:25:49.631198+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/HOKXIVCY3XOPJ5LZPTTMMNE7A4","json":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4.json","graph_json":"https://pith.science/api/pith-number/HOKXIVCY3XOPJ5LZPTTMMNE7A4/graph.json","events_json":"https://pith.science/api/pith-number/HOKXIVCY3XOPJ5LZPTTMMNE7A4/events.json","paper":"https://pith.science/paper/HOKXIVCY"},"agent_actions":{"view_html":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4","download_json":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4.json","view_paper":"https://pith.science/paper/HOKXIVCY","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=math/0001190&json=true","fetch_graph":"https://pith.science/api/pith-number/HOKXIVCY3XOPJ5LZPTTMMNE7A4/graph.json","fetch_events":"https://pith.science/api/pith-number/HOKXIVCY3XOPJ5LZPTTMMNE7A4/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4/action/storage_attestation","attest_author":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4/action/author_attestation","sign_citation":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4/action/citation_signature","submit_replication":"https://pith.science/pith/HOKXIVCY3XOPJ5LZPTTMMNE7A4/action/replication_record"}},"created_at":"2026-05-17T23:56:56.640726+00:00","updated_at":"2026-05-17T23:56:56.640726+00:00"}