# Fermat's polygonal number theorem for repeated generalized polygonal numbers

## TL;DR

This paper extends Fermat's polygonal number theorem to include repeated generalized polygonal numbers, establishing minimal and optimal bounds for representing all positive integers.

## Contribution

It generalizes Fermat's theorem to repeated generalized polygonal numbers and determines minimal and optimal bounds for their representations.

## Key findings

- Derived minimal number of generalized m-gonal numbers for all positive integers
- Established bounds for representations with repeated generalized m-gonal numbers
- Generalized Fermat's theorem to broader classes of polygonal numbers

## Abstract

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.