I have an equation of the form $$Dx^2-2s.t.x+t^2=c^2$$ where $s$, $t$ and $x$ are positive integers and $c$ can be any positive and odd integer. Is there a method to recursively find the values of $x$ that satisfy this particular equation? Let us take the example of $D=13$, $s=3$ and $t=2$. Then the values of $x$ which form the solution space are $660$ and $78480$. Is there a way to find it without resorting to brute force calculations?
A rather special case of this would be to take $D=s^2+t^2$. I have this solution on this site, General method for determining if $Ax^2 + Bx + C$ is square, but how do I proceed in this case of the residue being an odd square?
