How to inverse this equation which can give the devisors values of numbers

Question

The roots of the following equation

$f(x) = sin^2(\pi*x) + sin^2(\pi*n/x)$

are the positive and negative devisor of ($n$)

For example, if we set ($n=7$) then ($x$) will have 4 real values {-7,-1,1,7} and so for all prime numbers.

To make sure try to set integers to ($n$) and draw the equation.

Now the question is:

Is there any way to inverse this equation

$sin^2(\pi*x) + sin^2(\pi*n/x) = 0$

and find $x = f(n)$ by using Lambert W-function or any other technique?

If it is found then we can check the given number $n$ whether it is a prime or not by only calculating the equation real roots.

Answer 1

$\sin(\pi x)=0\implies x=\{k:k\in\Bbb{Z}\}$

$\sin(\pi n/x)=0\implies n/x=\{m:m\in\Bbb{Z}\}\implies n=\{xm:x,m\in\Bbb{Z}\}$