{"paper":{"title":"Sums of four polygonal numbers with coefficients","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xiang-Zi Meng, Zhi-Wei Sun","submitted_at":"2016-08-08T15:32:59Z","abstract_excerpt":"Let $m\\ge3$ be an integer. The polygonal numbers of order $m+2$ are given by $p_{m+2}(n)=m\\binom n2+n$ $(n=0,1,2,\\ldots)$. A famous claim of Fermat proved by Cauchy asserts that each nonnegative integer is the sum of $m+2$ polygonal numbers of order $m+2$. For $(a,b)=(1,1),(2,2),(1,3),(2,4)$, we study whether any sufficiently large integer can be expressed as $$p_{m+2}(x_1) + p_{m+2}(x_2) + ap_{m+2}(x_3) + bp_{m+2}(x_4)$$ with $x_1,x_2,x_3,x_4$ nonnegative integers. We show that the answer is positive if $(a,b)\\in\\{(1,3),(2,4)\\}$, or $(a,b)=(1,1)\\ \\&\\ 4\\mid m$, or $(a,b)=(2,2)\\ \\&\\ m\\not\\equiv"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.02022","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"}