Actually, what I've said you is a part of program... sorry for not disclosing everything.
The mathematical analysis for the puzzle ends at finding number of positive integral factors of (n!)^2 [ read as N factorial square ] .. Its a number to second power.
So, the answer for number of positive solutions would be ( factors( n! ^ 2) + 1 ) / 2
but the issue I am facing is to find the prime factors .. As n! itself would be huge, n! ^ 2 would be "Can't Imagine".. ex: take n = 1235.. n! itself goes too big. [ These are the numbers I got to deal with. max range of n = 10^6 ].
So, how could I do that... Its pretty well understood that I can't solve in brute force way.
I can't even use external lib. - I mean I shouldn't