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:

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.

Also known as
This theorem is often referred to as Gauss's Eureka Theorem, from 's famous diary entry.