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Given odd, coprime integers a, b (a > 0), we consider the Diophantine equation ax 2 + b 2l = 4y n , x, y ∈ Z, l ∈ N, n odd prime, gcd(x, y) = 1. We completely solve the above Diophantine equation for a ∈ {7, 11, 19, 43, 67, 163}, and b a power of an odd prime, under the conditions 2 n−1 b l ≡ ±1(mod a) and gcd(n, b) = 1. For other square-free integers a > 3 and b a power of an odd prime, we prove that the above Diophantine equation has no solutions for all integers x, y with (gcd(x, y) = 1), l ∈ N and all odd primes n > 3, satisfying 2 n−1 b l ≡ ±1(mod a), gcd(n, b) = 1, and gcd(n, h(−a)) = 1, where h(−a) denotes the class number of the imaginary quadratic field Q( √ −a).

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