Universal sums of generalized pentagonal numbers

TL;DR

This paper classifies all proper universal sums of generalized pentagonal numbers and proves a criterion, called the pentagonal theorem of 109, for their universality based on representation of specific integers.

Contribution

It identifies exactly 234 proper universal sums of generalized pentagonal numbers and establishes the pentagonal theorem of 109 as a criterion for universality.

Findings

234 proper universal sums identified

Pentagonal theorem of 109 proved

Universality characterized by representation of 12 specific integers

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 $109$.