# Triangular Lucas Numbers

## Theorem

The only Lucas numbers which are also triangular are:

$1, 3, 5778$

## Proof

 $\displaystyle 1$ $=$ $\displaystyle \dfrac {1 \times 2} 2$ $\displaystyle 3$ $=$ $\displaystyle \dfrac {2 \times 3} 2$ $\displaystyle = 2 + 1$ $\displaystyle 5778$ $=$ $\displaystyle \dfrac {107 \times 108} 2$ $\displaystyle = 2207 + 3571$