{"paper":{"title":"Refining Lagrange's four-square theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2016-04-22T16:00:36Z","abstract_excerpt":"Lagrange's four-square theorem asserts that any $n\\in\\mathbb N=\\{0,1,2,\\ldots\\}$ can be written as the sum of four squares. This can be further refined in various ways. We show that any $n\\in\\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\\in\\mathbb Z$ such that $x+y+z$ (or $x+2y$, $x+y+2z$) is a square (or a cube). We also prove that any $n\\in\\mathbb N$ can be written as $x^2+y^2+z^2+w^2$ with $x,y,z,w\\in\\mathbb N$ such that $P(x,y,z)$ is a square, whenever $P(x,y,z)$ is among the polynomials \\begin{gather*} x,\\ 2x,\\ x-y,\\ 2x-2y,\\ a(x^2-y^2)\\ (a=1,2,3),\\ x^2-3y^2,\\ 3x^2-2y^2, \\\\x^"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1604.06723","kind":"arxiv","version":14},"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"}