{"paper":{"title":"The Sum of Four Squares Over Real Quadratic Number Fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Katherine Thompson","submitted_at":"2016-10-21T20:05:08Z","abstract_excerpt":"Well-known results of Lagrange and Jacobi prove that the every $m \\in \\mathbb N$ can be expressed as a sum of four integer squares, and the number $r(m)$ of such representations can be given by an explicit formula in $m$. In this paper, we prove that the only real quadratic number field for which the sum of four squares is universal is $\\mathbb Q(\\sqrt{5})$. We provide explicit formulas for $r(m)$ for $K= \\mathbb Q (\\sqrt{2})$ and $K= \\mathbb Q(\\sqrt{5})$. We then consider the theta series of the sum of four squares over any real quadratic number field, providing explicit upper and lower bound"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1610.06935","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"}