Diophantine equations and class number of imaginary quadratic fields

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and $μ_{i} ∈ {-1,1}(i = 1,2)$, and let $h(-2^{1-e}D)(e = 0 or 1)$ denote the class number of the imaginary quadratic field $ℚ(√(-2^{1-e}D))$. In this paper, we give the all-positive integer solutions of the Diophantine equation Ax² + μ₁B = K((Ay² + μ₂B)/K)ⁿ, 2 ∤ n, n > 1 and we prove that if D > 1, then $h(-2^{1-e}D) ≡ 0 (mod n)$, where D, and n satisfy $kⁿ - 2^{e+1} = Dx²$, x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.