Let a, b, x, y, m, n be positive integers. Many special cases of the diophantine equation axwhere (ax, by) = 1, a, b are square-free integers, y > 1, m is odd and n is an odd prime, have been considered in the last few years (see [1, 3,[7][8][9][10][11]). Le Maohua [4-6] studied this equation in full generality and proved that it has only a finite number of solutions (a, b, x, y, m, n) with n > 5. Following almost the same method as Le Maohua but using a recent result of Bilu, Hanrot and Voutier [2], we are able to prove:Theorem. The diophantine equationwhere a, b, x, y, k, n are positive integers such that (ax, b) = 1, a, b are squarefree integers, k ≥ 0, n is an odd prime, (n, h) = 1 where h is the class number of the field Q( √ −ab) and y > 1, has no solutions in (a, b, x, y, k, n) when n > 13 and has exactly six solutions for 7 ≤ n ≤ 13, given by (a, b, k, n, y) = (1, 7, 0, 13, 2), (1,7, 1,7, 2), (1, 19, 0,7, 5), (3, 5, 0, 7, 2), (5, 7, 1, 7, 3), (13, 3, 0,7, 4).Further if a = 1, n = 5, then (1) has exactly 2 solutions given by k = 0 and (b, y) = (7, 2), (11, 3).We are grateful to Professor Bilu for his valuable suggestions and also for providing us with a copy of [1]. He also informed us that similar results using similar approach have been obtained by Bugeaud [3]. Our paper was submitted independently although later than that of Bugeaud.We start by giving some important definitions.