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In this paper, we consider sums of generalized polygonal numbers withrepeats, generalizing Fermat's polygonal number theorem which was proven byCauchy. In particular, we obtain the minimal number of generalized $m$-gonalnumbers required to represent every positive integer and we furthermoregeneralize 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.

Fermat's polygonal number theorem for repeated generalized polygonal numbers