{"paper":{"title":"On $\\theta$-congruent numbers on real quadratic number fields","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Ali S. Janfada, Sajad Salami","submitted_at":"2014-12-10T11:10:21Z","abstract_excerpt":"Let ${\\mathbb K}={\\mathbb Q}(\\sqrt{m})$ be a real quadratic number field, where $m>1$ is a squarefree integer. Suppose that $0 < \\theta< \\pi $ has rational cosine, say $\\cos (\\theta)=s/r$ with $0< |s|<r$ and $\\gcd(r,s)=1$. A positive integer $n$ is called a $(\\mathbb K,\\theta)$-congruent number if there is a triangle, called the $(\\mathbb K,\\theta, n)$-triangles, with sides in $\\mathbb K$ having $\\theta$ as an angle and $n\\alpha_\\theta$ as area, where ${\\alpha_\\theta}=\\sqrt{r^2-s^2}$. Consider the $(\\mathbb K,\\theta)$-congruent number elliptic curve $E_{n,\\theta}: y^2=x(x+(r+s)n)(x-(r-s)n)$ de"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.3258","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"}