Start with ((2M – N)/(2M + N)), ignoring everything else.

Let n = ((2M – N)/(2M + N)), requiring n to be positive, since it is a square root leading to a prime solution. M and N are required to be positive anyway.

But if so, then N = -(2M(n-1)))/(n+1), which cannot be positive. So it looks as though the problem falls at that hurdle.

I'm not seeing how to get past this to the rest of the problem.