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Diophantine equations and class numbers of real quadratic fields

Xiaolei Dong; Zhenfu Cao — 2001

Acta Arithmetica

An application of a lower bound for linear forms in two logarithms to the Terai-Jeśmanowicz conjecture

Zhenfu Cao; Xiaolei Dong — 2003

Acta Arithmetica

Diophantine equations and class number of imaginary quadratic fields

Zhenfu Cao; Xiaolei Dong — 2000

Discussiones Mathematicae - General Algebra and Applications

Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and $μ_{i} \in - 1, 1 (i = 1, 2)$ , and let $h (- 2^{1 - e} D) (e = 0 o r 1)$ denote the class number of the imaginary quadratic field $ℚ (\sqrt (- 2^{1 - e} D))$ . 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 $h (- 2^{1 - e} D) \equiv 0 (m o d n)$ , where D, and n satisfy $k ⁿ - 2^{e + 1} = D x ²$ , 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

Xiaolei Dong; W. C. Shiu; C. I. Chu; Zhenfu Cao — 2007

Acta Arithmetica

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Currently displaying 1 – 4 of 4

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

The simultaneous Pell equations y²-Dz² = 1 and x²-2Dz² = 1