Journal of the
Korean Mathematical Society JKMS

For an integer $m \ge 3$, every integer of the form $p_m(x)=\frac{(m-2)x^2-(m-4)x}2$ with $x \in \mathbb Z$ is said to be a generalized $m$-gonal number. Let $a\le b\le c$ and $k$ be positive integers. The quadruple $(k,a,b,c)$ is said to be {\it universal} if for every nonnegative integer $n$ there exist integers $x,y,z$ such that $n=ap_k(x)+bp_k(y)+cp_k(z)$. Sun proved in [16] that, when $k=5$ or $k \ge 7$, there are only $20$ candidates for universal quadruples, which he listed explicitly and which all involve only the case of pentagonal numbers ($k=5$). He verified that six of the candidates are in fact universal and conjectured that the remaining ones are as well. In a subsequent paper [3], Ge and Sun established universality for all but seven of the remaining candidates, leaving only $(5,1,1,t)$ for $t=6,8,9,10$, $(5,1,2,8)$ and $(5,1,3,s)$ for $s=7, 8$ as candidates. In this article, we prove that the remaining seven quadruples given above are, in fact, universal.

Keywords: generalized polygonal numbers, ternary universal sums

MSC numbers: Primary 11E12, 11E20