Find $m,n \in \mathbb{N}$, such that $\sqrt{\frac{m(m+1)}{n(n+1)}}$ is a rational number

Question

Find $m,n \in \mathbb{N}$, $m \neq n$, such that $N = \sqrt{\frac{m(m+1)}{n(n+1)}}$ is a rational number.

Answer 1

There are infinitely solutions because there are infinitely many square triangular numbers.

Just choose $m$ and $n$ from the sequence OEIS/A001108: $$ 1,8,49,288,1681,9800,57121,332928,1940449,11309768,65918161,384199200,\dots $$