Universal sums of generalized pentagonal numbers

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 $α_1,\dots,α_k$, a sum $Φ_{α_1,\dots,α_k}(x_1,x_2,\dots,x_k)=α_1P_5(x_1)+α_2P_5(x_2)+\cdots+α_kP_5(x_k)$ of generalized pentagonal numbers is called universal if $Φ_{α_1,\dots,α_k}(x_1,x_2,\dots,x_k)=N$ 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 $Φ_{α_1,\dots,α_k}(x_1,x_2,\dots,x_k)$ is universal if and only if it represents the integers $1, 3, 8, 9, 11, 18, 19, 25, 27, 43, 98$, and $109$.

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