On the Diophantine equation ax2 + by2 = ckn

In this paper we investigate the solvability and the representation of the solutions of the equation ax2 +by2 = ckn. We extend and improve many known results. In particular, we completely solve the equation (a ± 1)x2 + (3a Ŧ 1) = 4an, 2 ∤ n.