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A note on the Diophantine equation $a^x + b^y = c^z$

100%

Cao Z.

Acta Arithmetica

1999

tom 91

nr 1

85-93

Diophantine equations and class numbers of real quadratic fields

63%

Dong X. , Cao Z.

Acta Arithmetica

2001

tom 97

nr 4

313-328

Some classes of Diophantine equations connected with McFarland's and Ma's conjectures

63%

Cao Z. , Grytczuk A.

Discussiones Mathematicae - General Algebra and Applications

2000

tom 20

nr 2

193-198

In this paper we consider some special classes of Diophantine equations connected with McFarland's and Ma's conjectures about difference sets in abelian groups and we obtain an extension of known results.

Diophantine equations and class number of imaginary quadratic fields

63%

Cao Z. , Dong X.

Discussiones Mathematicae - General Algebra and Applications

2000

tom 20

nr 2

199-206

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 $h(-2^{1-e}D)(e = 0 or 1)$ denote the class number of the imaginary quadratic field $ℚ(√(-2^{1-e}D))$. 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 $h(-2^{1-e}D) ≡ 0 (mod n)$, 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.

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

63%

Cao Z. , Dong X.

Acta Arithmetica

2003

tom 110

nr 2

153-164

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

51%

Dong X. , Shiu W. , Chu C. , Cao Z.

nr 2

115-123