Find integers that satisfy $a^2b -2N = -c^2$ where $a,b,c,N$ are positive integers, and $N$ is a big integer that I am trying to factor.
One direction that I had is to multiply $N$ by a small integer $k$ because then I can say:
$$\left(\frac{n+k}{2}\right)^2-\left(\frac{n-k}{2}\right)^2=Nk$$
It's pretty close to $a^2b +c^2 = 2N$, but then I don't see how it can help. Also $k$ must be an odd number, while I really want it to be 2.
Solving it will help me with my factorization idea here.
Here is an example of such numbers.
- $N = 3161= 29 * 109$
- $a = 9$
- $b = 78$
- $c = 2$