A recent puzzle asked for the smallest number that can be written as the difference of 2 squares in 4 ways - very interesting but not that challenging.

What is the smallest number that can be written as the (positive) difference of 2 squares in exactly 997 ways?

I am totally flabbergasted.

Can it be solved analytically?

Is 997 aspecial number or your solution works for any n ?

A tiny hint (i.e. where to start)  will be appreciated..