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 ${\Phi }_{{\alpha }_1,\dots ,{\alpha }_k}(x_1,x_2,\dots ,x_k)={\alpha }_1{P}_5(x_1)+{\alpha }_2{P}_5(x_2)+\cdots +{\alpha }_k{P}_5(x_k)$ 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 $109$.

14 pages