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 $(m^2+1, m)$.
If $d\in\Bbb N$ is not a square, and $$x^2-dy^2=k\tag1$$ has some integer solution for $k\ne 0$ then $(1)$ has infinitely many integer solutions. This is because Pell's equation $$x^2-dy^2=1\tag2$$ has infinitely many integer solutions, and if $(x_1,y_1)$ is a solution to $(1)$ and $(x_2,y_2)$ is a solution to $(2)$ then $(x_1x_2+dy_1y_2,x_1y_2+y_1x_2)$ is also a solution to $(1)$.