# Polygonal Numbers

> EYPHKA! num = (triangle) + (triangle) + (triangle)
>
> Carl Friedrich Gauss


## Introduction

### Triangular numbers

T0 = 0

.

T1 = 1

 .
. .

T2 = 3

  .
 . .
. . .

T3 = 6

(...)

### Square numbers

S0 = 0

.

S1 = 1

..
..

S2 = 4

...
...
...

S3 = 9

(...)

### General case

The formula for the n-th s-gonal number is

P(s,n) = [(s-2)*n^2 - (s-4)*n] / 2


## Fermat polygonal number theorem

Thm. Every positive integer is a sum of at most n n-gonal numbers.

For example,

27
= 21 + 6 + 0 (Triangular)
= 16 + 9 + 1 + 1 (Square)
= 22 + 5 + 0 + 0 + 0 (Pentagonal)
= 15 + 6 + 6 + 0 + 0 + 0 (Hexagonal)
...

n=4 case is especially famous. It is called Lagrange's four-square
theorem.