{"paper":{"title":"Restricted sums of four squares","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2017-01-20T17:34:21Z","abstract_excerpt":"We refine Lagrange's four-square theorem in new ways by imposing some restrictions involving powers of two (including $1$). For example, we show that each $n=1,2,3,\\ldots$ can be written as $x^2+y^2+z^2+w^2$ $(x,y,z,w\\in\\mathbb N=\\{0,1,2,\\ldots\\})$ with $|x+y-z|\\in\\{4^k:\\ k\\in\\mathbb N\\}$ (or $|2x-y|\\in\\{4^k:\\ k\\in\\mathbb N\\}$, or $x+y-z\\in\\{\\pm 8^k:\\ k\\in\\mathbb N\\}\\cup\\{0\\}\\subseteq\\{t^3:\\ t\\in\\mathbb Z\\}$), and that we can write any positive integer as $x^2+y^2+z^2+w^2$ $(x,y,z,w\\in\\mathbb Z)$ with $x+y+2z$ (or $x+2y+2z$) a power of four. We also prove that any $n\\in\\mathbb N$ can be writte"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.05868","kind":"arxiv","version":10},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1701.05868/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}