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 Odd Squares, $8 n + 3$ is the sum of $3$ odd squares.

So:

$\ds \forall n \in \Z_{\ge 0}: \, $

$\ds 8 n + 3$

$=$

$\ds \paren {2 x + 1}^2 + \paren {2 y + 1}^2 + \paren {2 z + 1}^2$

for some $x, y, z \in \Z_{\ge 0}$

$\ds $

$=$

$\ds 4 x^2 + 4 x + 4 y^2 + 4 y + 4 z^2 + 4 z + 3$

$\ds $

$=$

$\ds 4 \paren {x \paren {x + 1} + y \paren {y + 1} + z \paren {z + 1} } + 3$

$\ds \leadsto \ \ $

$\ds n$

$=$

$\ds \frac {x \paren {x + 1} } 2 + \frac {y \paren {y + 1} } 2 + \frac {z \paren {z + 1} } 2$

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 .

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): $3$