  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.
