{"paper":{"title":"Natural numbers represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor+\\lfloor z^2/c\\rfloor$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2015-04-06T15:56:21Z","abstract_excerpt":"Let $a,b,c$ be positive integers. It is known that there are infinitely many positive integers not representated by $ax^2+by^2+cz^2$ with $x,y,z\\in\\mathbb Z$. In contrast, we conjecture that any natural number is represented by $\\lfloor x^2/a\\rfloor+\\lfloor y^2/b\\rfloor +\\lfloor z^2/c\\rfloor$ with $x,y,z\\in\\mathbb Z$ if $(a,b,c)\\not=(1,1,1),(2,2,2)$, and that any natural number is represented by $\\lfloor T_x/a\\rfloor+\\lfloor T_y/b\\rfloor+\\lfloor T_z/c\\rfloor$ with $x,y,z\\in\\mathbb Z$, where $T_x$ denotes the triangular number $x(x+1)/2$. We confirm this general conjecture in some special cases"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.01608","kind":"arxiv","version":8},"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"}