In this paper, we consider sums of generalized polygonal numbers with repeats, generalizing Fermat's polygonal number theorem which was proven by Cauchy. In particular, we obtain the minimal number of generalized $m$-gonal numbers required to represent every positive integer and we furthermore generalize this result to obtain optimal bounds when many of the generalized $m$-gonal numbers are repeated $r$ times, where $r\in\mathbb{N}$ is fixed.

中文翻译：

---

重复广义多边形数的费马多边形数定理

在本文中，我们考虑具有重复的广义多边形数的和，推广由柯西证明的费马多边形数定理。特别是，我们获得了表示每个正整数所需的最小广义 $m$-gonal 数，并且当许多广义 $m$-gonal 数重复 $r$ 次时，我们进一步推广该结果以获得最佳边界，其中 $r\in\mathbb{N}$ 是固定的。