{"paper":{"title":"Periodic continued fractions and elliptic curves over quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Mohammad Sadek","submitted_at":"2014-11-22T22:17:49Z","abstract_excerpt":"Let $f(x)$ be a square free quartic polynomial defined over a quadratic field $K$ such that its leading coefficient is a square. If the continued fraction expansion of $\\displaystyle \\sqrt{f(x)}$ is periodic, then its period $n$ lies in the set \\[\\{1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,17,18,22,26,30,34\\}.\\] We write explicitly all such polynomials for which the period $n$ occurs over $K$ but not over $\\Q$ and $\\displaystyle n\\not\\in\\{ 13,15,17\\}$. Moreover we give necessary and sufficient conditions for the existence of such continued fraction expansions with period $13, 15$ or $17$ over $K$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1411.6174","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"}