It has been proved that if p is an odd prime, 1$"> y > 1 , k ≥ 0 , n is an integer greater than or equal to 4 , ( n , 3 h ) = 1 where h is the class number of the field Q ( − p ) , then the equation x 2 + p 2 k + 1 = 4 y n has exactly five families of solution in the positive integers x , y . It is further proved that when n = 3 and p = 3 a 2 ± 4 , then it has a unique solution k = 0 , y = a 2 ± 1 .