I wanted to solve this problem. You have to prove that there are infinitely many square numbers of the form $(50^m - 50^n)$ (and no square numbers of the form $(2020^m + 2020^n)$ with $m$ and $n$ being positive whole numbers.
I tried many approaches but just couldn‘t manage to do it. For example: if you factor out $50^n$ you get $50^n(50^{(m-n)}-1)$ and that means, that if $a = m-n$, then $50^a - 1$ is a square as well, but I couldn't prove that either.
I just need an approach to see how I could go about solving such a problem. Thanks
$$50^{2k+1}-50^{2k}=(7\cdot 50^k)^2$$