Integer is Sum of Three Triangular Numbers

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Theorem

Let $n$ be a positive integer.

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


Proof

From Integer as Sum of Three Odd Squares, $8 n + 3$ is 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