{"paper":{"title":"On some universal sums of generalized polygonal numbers","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Fan Ge, Zhi-Wei Sun","submitted_at":"2009-06-15T03:47:06Z","abstract_excerpt":"For $m=3,4,\\ldots$ those $p_m(x)=(m-2)x(x-1)/2+x$ with $x\\in\\mathbb Z$ are called generalized $m$-gonal numbers. Sun [13] studied for what values of positive integers $a,b,c$ the sum $ap_5+bp_5+cp_5$ is universal over $\\mathbb Z$ (i.e., any $n\\in\\mathbb N=\\{0,1,2,\\ldots\\}$ has the form $ap_5(x)+bp_5(y)+cp_5(z)$ with $x,y,z\\in\\mathbb Z$). We prove that $p_5+bp_5+3p_5\\,(b=1,2,3,4,9)$ and $p_5+2p_5+6p_5$ are universal over $\\mathbb Z$, as conjectured by Sun. Sun also conjectured that any $n\\in\\mathbb N$ can be written as $p_3(x)+p_5(y)+p_{11}(z)$ and $3p_3(x)+p_5(y)+p_7(z)$ with $x,y,z\\in\\mathbb "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0906.2450","kind":"arxiv","version":3},"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"}