I want to show that for $d= m^2+2$ the equation
$$x^2-dy^2 = -2$$
has infinitetly many integer solutions. By trying out one can see that $x=\pm m, y=\pm 1$ are solutions, but how do we know that there exist infinitely many solutions?
I have already studied the similar equation $x^2 -dy^2=1$ and shown that for that a fundamental solution is given by $(m^2+1, m)$.