Fermat's Polygonal Number Theorem

Fermat's polygonal number theorem states that every positive integer can be written as the sum of at most $n$ $n$-gon numbers. This implies that a positive integer can be written as the sum of at most 3 triangular numbers, 4 square numbers, and so on.

• $17=1+6+10$ (triangular numbers)
• $17=1+16$ (square numbers)
• $17=5+12$ (pentagonal numbers)

The case for $n=3$ was proved by Gauss and the proof is given below.

From Lagrange's three-square theorem, we know that a number $r$ of the form $8n+7$ cannot be represented as the sum of 3 square numbers. This implies that numbers of the form $8n+3$ can be written as the sum of 3 squares.

The squares mod 8 are $(0,1)$ and $(4).$

For $r$ to be the sum of three squares, the squares need to be congruent to $1 \pmod 8:$

$$\begin{aligned} r&=(2x+1)^2+(2y+1)^2+(2z+1)^2\\ 8n+3&=4x^2+4x+4y^2+4y+4z^2+4z+3\\ 8n&=4x(x+1)+4y(y+1)+4z(z+1)\\ n&=\frac{x(x+1)}{2}+\frac{y(y+1)}{2}+\frac{z(z+1)}{2}. \end{aligned}$$

All triangular numbers are of the form $\frac{a(a+1)}{2}.$ Thus, $n$ can be written as the sum of three triangular numbers. $_\square$