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:

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.