Diophantine equations and class numbers of real quadratic fields
An application of a lower bound for linear forms in two logarithms to the Terai-Jeśmanowicz conjecture
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 , and let denote the class number of the imaginary quadratic field . 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 , where D, and n satisfy , x ∈ ℕ, 2 ∤ n, n > 1. The results are valuable for the realization of quadratic field cryptosystem.
The simultaneous Pell equations y²-Dz² = 1 and x²-2Dz² = 1
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