{"paper":{"title":"Proof of a conjecture of Guy on class numbers","license":"http://creativecommons.org/licenses/by-nc-sa/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Allen Yuan, Benjamin Gunby, Lynn Chua, Soohyun Park","submitted_at":"2014-07-11T19:45:31Z","abstract_excerpt":"It is well known that for any prime $p\\equiv 3$ (mod $4$), the class numbers of the quadratic fields $\\mathbb{Q}(\\sqrt{p})$ and $\\mathbb{Q}(\\sqrt{-p})$, $h(p)$ and $h(-p)$ respectively, are odd. It is natural to ask whether there is a formula for $h(p)/h(-p)$ modulo powers of $2$. We show the formula $h(p) \\equiv h(-p) m(p)$ (mod $16$), where $m(p)$ is an integer defined using the \"negative\" continued fraction expansion of $\\sqrt{p}$. Our result solves a conjecture of Richard Guy."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1407.3261","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"}