Universal sums of generalized pentagonal numbers

Abstract

For an integer $x$, an integer of the form $P_5(x)=\frac{3x^2-x}2$ is called a generalized pentagonal number. For positive integers $\alpha_1,\dots,\alpha_k$, a sum $\Phi_{\alpha_1,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=\alpha_1P_5(x_1)+\alpha_2P_5(x_2)+\cdots+\alpha_kP_5(x_k)$ of generalized pentagonal numbers is called universal if $\Phi_{\alpha_1,\dots,\alpha_k}(x_1,x_2,\dots,x_k)=N$ has an integer solution $(x_1,x_2,\dots,x_k) \in \mathbb Z^k$ for any non-negative integer $N$. In this article, we prove that there are exactly $234$ proper universal sums of generalized pentagonal numbers. Furthermore, the "pentagonal theorem of $109$" is proved, which states that an arbitrary sum $\Phi_{\alpha_1,\dots,\alpha_k}(x_1,x_2,\dots,x_k)$ is universal if and only if it represents the integers $1, 3, 8, 9, 11, 18, 19, 25, 27, 43, 98$, and $109$.