Let $\displaystyle a_{n}\subset\mathbb{N} $ define as

$\displaystyle a_{1}= 36\\ a_{2}= 361\\ a_{3}= 3611\\ a_{4}= 36111\\ a_{5}= 361111\\ \cdots\\ a_{n}= 36111\cdots 11\\ \cdots $

When $\displaystyle a_{m}* a_{n}\; (m < n) $ is square number ?