Geodesic
Diophantine equations and class number of imaginary quadratic fields
Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 2, pp. 199-206

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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-eD)(e = 0 or 1) denote the class number of the imaginary quadratic field ℚ(√(-2^1-eD)). 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-eD) ≡ 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.
Keywords: Diophantine equation, imaginary quadratic field, class number, cryptographic problem 
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Cao, Zhenfu; Dong, Xiaolei. Diophantine equations and class number of imaginary quadratic fields. Discussiones Mathematicae. General Algebra and Applications, Tome 20 (2000) no. 2, pp. 199-206. http://geodesic.mathdoc.fr/item/DMGAA_2000_20_2_a4/