Diophantine Equations and Class Numbers of Imaginary Quadratic Fields

Let A,D, K, k ∈ N with D square free and 2 | /k, B = 1, 2 or 4 and μi ∈ {−1, 1}(i = 1, 2), and let h(−21−eD)(e = 0 or 1) denote the class number of the imaginary quadratic field Q( √−21−eD). In this paper, we give the all-positive integer solutions of the Diophantine equation Ax +μ1B = K ( (Ay +μ2B)/K )n , 2 | / n, n > 1 and we prove that if D > 1, then h(−21−eD) ≡ 0(mod n), where D, and n satisfy k − 2 = Dx, x ∈ N, 2 | / n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.