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 ${\displaystyle {\mu }_i\in \{-1,1\}(i=1,2)}$, and let ${\displaystyle h(-2^{1-e}D)(e=0\text{or}1)}$ denote the class number of the imaginary quadratic field ${\displaystyle \mathbb{Q}(\surd (-2^{1-e}D))}$. 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 ${\displaystyle h(-2^{1-e}D)\equiv 0(modn)}$, where D, and n satisfy ${\displaystyle kⁿ-2^{e+1}=Dx²}$, x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.