# Triangular Number whose Square is Triangular

## Theorem

The only triangular number with less than $660$ digits, whose square is also triangular, is $6$.

## Proof

We have that:

${T_3}^2 = 6^2 = 36 = 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8$

To establish that it is the only one yet known can be established by an exhaustive search.

$\blacksquare$