Prove that whenever the equation $x^2 - dy^2 = c$ is solvable, then it has infinitely many solutions.
I consider that, if $u$ and $v$ satisfy $x^2 -dy^2 = c$ and then $r$ and $s$ satisfy $x^2 -cy^2 = 1$, then $$(ur \pm dvs)^2 - d(us \pm vr)^2 = (u^2 - dv^2)(r^2 - ds^2) = c\;.$$ But, still I failed to complete the proof. I am requesting members to spare some time for this. Thanks in advnace.