{"paper":{"title":"On $x(ax+1)+y(by+1)+z(cz+1)$ and $x(ax+b)+y(ay+c)+z(az+d)$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Zhi-Wei Sun","submitted_at":"2015-05-14T16:00:43Z","abstract_excerpt":"In this paper we first investigate for what positive integers $a,b,c$ every nonnegative integer $n$ can be represented as $x(ax+1)+y(by+1)+z(cz+1)$ with $x,y,z$ integers. We show that $(a,b,c)$ can be either of the following seven triples: $$(1,2,3),\\ (1,2,4),\\ (1,2,5),\\ (2,2,4),\\ (2,2,5),\\ (2,3,3),\\ (2,3,4),$$ and conjecture that any triple $(a,b,c)$ among $$(2,2,6),\\ (2,3,5),\\ (2,3,7),\\ (2,3,8),\\ (2,3,9),\\ (2,3,10)$$ also has the desired property. For integers $0\\le b\\le c\\le d\\le a$ with $a>2$, we prove that any nonnegative integer can be represented as $x(ax+b)+y(ay+c)+z(az+d)$ with $x,y,z"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.03679","kind":"arxiv","version":4},"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"}