# Find number of integral solution.

Find the no of positive integral solutions for the equations

$$\frac{1}{x}+\frac{1}{y}=\frac{1}{N!}$$

I just come across this problem. I know we can resolve the problem to

$$(x-N!)(y-N!)=(N!)^2$$

so number of possible divisor for this is

$$(2e_1+1)(2e_2+1)\dotsm(2e_n+1)$$

Where $e_1,e_2,\dots,e_n$ are exponent power of primes factor.

Now how can I find the number of positive integral solution for a very large value?

I think you are doing fine. Note that the exponent of $p$ in $N!$ can be obtained as $\lfloor N/p\rfloor + \lfloor N/p^2\rfloor + \lfloor N/p^3\rfloor +\ldots$. –  Hagen von Eitzen Nov 29 '13 at 21:51