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 proven, 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.

中文翻译：

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广义五角数的通用和

对于整数x，形式为\（P_5（x）= \ frac {3x ^ 2-x} {2} \）的整数称为广义五边形数。对于正整数\（\ alpha _1，\ dots，\ alpha _k \），总和\（\ Phi _ {\ alpha _1，\ dots，\ alpha _k}（x_1，x_2，\ dots，x_k）= \ alpha如果通用\（\ Phi _ {\ alpha _1，\ dots，\ alpha _k}（x_1 ）被称为通用五角形数的_1P_5（x_1）+ \ alpha _2P_5（x_2）+ \ cdots + \ alpha _kP_5（x_k）\）被称为通用五角形，x_2，\ dots，x_k）= N \）具有整数解\（（x_1，x_2，\ dots，x_k）\ in {\ mathbb {Z}} ^ k \）对于任何非负整数N。在本文中，我们证明了精确存在234个适当的广义五角形数的通用和。此外，证明了“ 109的五边形定理”，它指出，任意和\（\ Phi _ {\ alpha _1，\ dots，\ alpha _k}（x_1，x_2，\ dots，x_k）\）是通用的并且仅当它表示整数1、3、8、9、11、18、19、25、27、43、98和109时。