Let d be a fixed positive integer with d > 3 is square-free, and let h ( − d ) denote the class number of the imaginary quadratic field ℚ ( − d ) . Further, let p and q be odd primes such that p > 3 , p ≠ q and p ∤ h ( − d ) . In this paper, we give a sufficient and necessary condition for the Lebesgue-Nagell equation (∗) d x 2 + p 2 m q 2 n = 4 y p to have positive integer solutions ( x , y , m , n ) with gcd ( x , y ) = 1 . It can be seen from this condition that if q ≢ ± 1 ( 𝑚𝑜𝑑 2 p ) , then (∗) has no positive integer solutions ( x , y , m , n ) with gcd ( x , y ) = 1 .
Le et al. (Thu,) studied this question.