# Integer is Sum of Three Triangular Numbers

## Theorem

Let $n$ be a positive integer.

Then $n$ is the sum of $3$ triangular numbers.

## Proof

From Integer as Sum of Three Squares, every positive integer not of the form $4^n \paren {8 m + 7}$ can be expressed as the sum of three squares.

Hence every positive integer $r$ such that $r \equiv 3 \pmod 8$ can likewise be expressed as the sum of three squares.

From Square Modulo 8, the squares modulo $8$ are $0, 1$ and $4$.

Thus for $r$ to be the sum of three squares, each of those squares needs to be congruent modulo $8$ to $1$.

Thus each square is odd, and $r$ can be expressed in the form $8 n + 3$ as the sum of $3$ odd squares.

So:

\(\, \displaystyle \forall n \in \Z_{\ge 0}: \, \) | \(\displaystyle 8 n + 3\) | \(=\) | \(\displaystyle \paren {2 x + 1}^2 + \paren {2 y + 1}^2 + \paren {2 z + 1}^2\) | for some $x, y, z \in \Z_{\ge 0}$ | |||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 4 x^2 + 4 x + 4 y^2 + 4 y + 4 z^2 + 4 z + 3\) | |||||||||||

\(\displaystyle \) | \(=\) | \(\displaystyle 4 \paren {x \paren {x + 1} + y \paren {y + 1} + z \paren {z + 1} } + 3\) | |||||||||||

\(\displaystyle \leadsto \ \ \) | \(\displaystyle n\) | \(=\) | \(\displaystyle \frac {x \paren {x + 1} } 2 + \frac {y \paren {y + 1} } 2 + \frac {z \paren {z + 1} } 2\) | subtracting $3$ and dividing both sides by $8$ |

By Closed Form for Triangular Numbers, each of $\dfrac {x \paren {x + 1} } 2$, $\dfrac {y \paren {y + 1} } 2$ and $\dfrac {z \paren {z + 1} } 2$ are triangular numbers.

$\blacksquare$

## Also known as

This theorem is often referred to as **Gauss's Eureka Theorem**, from Carl Friedrich Gauss's famous diary entry.

## Historical Note

Carl Friedrich Gauss proved that every Integer is Sum of Three Triangular Numbers.

The $18$th entry in his diary, dated $10$th July $1796$, made when he was $19$ years old, reads:

- $**\Epsilon\Upsilon\Rho\Eta\Kappa\Alpha \quad \text{num} = \Delta + \Delta + \Delta.$

## Sources

- 1986: David Wells:
*Curious and Interesting Numbers*... (previous) ... (next): $3$ - 1997: David Wells:
*Curious and Interesting Numbers*(2nd ed.) ... (previous) ... (next): $3$