### 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.

### License

BSD License

### Topics

- Mathematics/Number theory
- Mathematics/Combinatorics

### Related publications

- Nathanson, M. B. (1996). Additive Number Theory. In Graduate Texts in Mathematics. Springer New York. https://doi.org/10.1007/978-1-4757-3845-2

### Session Polygonal_Number_Theorem

- Polygonal_Number_Theorem_Lemmas
- Polygonal_Number_Theorem_Gauss
- Polygonal_Number_Theorem_Cauchy
- Polygonal_Number_Theorem_Legendre

### Depends on

- Three Squares Theorem

### Auto-related entries

- Wieferich–Kempner Theorem

## Cite

×

Polygonal_Number_Theorem-AFP

    @article{Polygonal_Number_Theorem-AFP,
      author  = {Kevin Lee and Zhengkun Ye and Angeliki Koutsoukou-Argyraki},
      title   = {Polygonal Number Theorem},
      journal = {Archive of Formal Proofs},
      month   = {August},
      year    = {2023},
      note    = {\url{https://isa-afp.org/entries/Polygonal_Number_Theorem.html},
                 Formal proof development},
      ISSN    = {2150-914x},
    }

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