{"paper":{"title":"Extended Lagrange's four-square theorem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Jes\\'us Lacalle, Laura N. Gatti","submitted_at":"2018-05-11T12:18:50Z","abstract_excerpt":"Lagrange's four-square theorem states that every natural number $n$ can be represented as the sum of four integer squares: $n=x_1^2+x_2^2+x_3^2+x_4^2$. Ramanujan generalized Lagrange's result by providing, up to equivalence, all $54$ quadratic forms $ax_1^2+bx_2^2+cx_3^2+dx_4^2$ that represent all positive integers. In this article, we prove the following extension of Lagrange's theorem: given a prime number $p$ and $v_1\\in Z^4$, $\\dots$, $v_k\\in Z^4$, $1\\leq k\\leq 3$, such that $\\|v_i\\|^2=p$ for all $1\\leq i\\leq k$ and $\\langle v_i|v_j\\rangle=0$ for all $1\\leq i<j\\leq k$, then there exists $v"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.04353","kind":"arxiv","version":1},"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"}