Let h denote the class number of the quadratic field ℚ ( − A ) for a square free odd integer 1$"> A > 1 , and suppose that 2$"> n > 2 is an odd integer with ( n , h ) = 1 and 1$"> m > 1 . In this paper, it is proved that the equation of the title has no solution in positive integers x and y if n has any prime factor congruent to 1 modulo 4. If n has no such factor it is proved that there exists at most one solution with x and y odd. The case n = 3 is solved completely. A result of E. Brown for A = 3 is improved and generalized to the case where A is a prime ≢ 7 ( mod 8 ) .