# Polygonal Number Theorem

August 10, 2023

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

### Abstract

We formalize the proofs of Cauchy's and Legendre's Polygonal Number Theorems given in Melvyn B. Nathanson's book "Additive Number Theory: The Classical Bases".

For $m \geq 1$, the $k$-th polygonal number of order $m+2$ is defined to be $p_m(k)=\frac{mk(k-1)}{2}+k$. The theorems state that:

1. If $m \ge 4$ and $N \geq 108m$, then $N$ can be written as the sum of $m+1$ polygonal numbers of order $m+2$, at most four of which are different from $0$ or $1$. If $N \geq 324$, then $N$ can be written as the sum of five pentagonal numbers, at least one of which is $0$ or $1$.

2. Let $m \geq 3$ and $N \geq 28m^3$. If $m$ is odd, then $N$ is the sum of four polygonal numbers of order $m+2$. If $m$ is even, then $N$ is the sum of five polygonal numbers of order $m+2$, at least one of which is $0$ or $1$.

We also formalize the proof of Gauss's theorem which states that every non-negative integer is the sum of three triangular numbers.