On metric theory of Diophantine approximation for complex numbers

Zhengyu Chen

Displaying similar documents to “On metric theory of Diophantine approximation for complex numbers”

Diophantine equations and class number of imaginary quadratic fields

Zhenfu Cao, Xiaolei Dong (2000)

Discussiones Mathematicae - General Algebra and Applications

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Let A, D, K, k ∈ ℕ with D square free and 2 ∤ k,B = 1,2 or 4 and $μ_{i} \in - 1, 1 (i = 1, 2)$ , 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.

On the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$

Florian Luca, Alain Togbé (2009)

Colloquium Mathematicae

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We find all the solutions of the Diophantine equation $x ² + 2^{α} 13^{β} = y ⁿ$ in positive integers x,y,α,β,n ≥ 3 with x and y coprime.

Further remarks on Diophantine quintuples

Mihai Cipu (2015)

Acta Arithmetica

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A set of m positive integers with the property that the product of any two of them is the predecessor of a perfect square is called a Diophantine m-tuple. Much work has been done attempting to prove that there exist no Diophantine quintuples. In this paper we give stringent conditions that should be met by a putative Diophantine quintuple. Among others, we show that any Diophantine quintuple a,b,c,d,e with a < b < c < d < e $s a t i s f i e s$ d < 1.55·1072 $a n d$ b < 6.21·1035 $w h e n 4 a < b, w h i l e f o r b < 4 a o n e h a s e i t h e r$ c = a + b + 2√(ab+1)...

The Diophantine equation ${(b n)}^{x} + {(2 n)}^{y} = {((b + 2) n)}^{z}$

Min Tang, Quan-Hui Yang (2013)

Colloquium Mathematicae

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Recently, Miyazaki and Togbé proved that for any fixed odd integer b ≥ 5 with b ≠ 89, the Diophantine equation $b^{x} + 2^{y} = {(b + 2)}^{z}$ has only the solution (x,y,z) = (1,1,1). We give an extension of this result.

On the Diophantine equation $x^{p} + y^{2 p} = z^{2}$

A. Rotkiewicz, A. Schinzel (1987)

Colloquium Mathematicae

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On $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$

Susil Kumar Jena (2015)

Communications in Mathematics

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The two related Diophantine equations: $X_{1}^{4} + 4 X_{2}^{4} = X_{3}^{8} + 4 X_{4}^{8}$ and $Y_{1}^{4} = Y_{2}^{4} + Y_{3}^{4} + 4 Y_{4}^{4}$ , have infinitely many nontrivial, primitive integral solutions. We give two parametric solutions, one for each of these equations.

On the system of Diophantine equations $a^{2} + b^{2} = {(m^{2} + 1)}^{r}$ and $a^{x} + b^{y} = {(m^{2} + 1)}^{z}$

Florian Luca (2012)

Acta Arithmetica

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The Diophantine Equation X³ = u+v over Real Quadratic Fields

Takaaki Kagawa (2011)

Bulletin of the Polish Academy of Sciences. Mathematics

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Let k be a real quadratic field and let $_{k}$ and $_{k}^{\times}$ be the ring of integers and the group of units, respectively. A method of solving the Diophantine equation X³ = u+v ( $X \in_{k}$ , $u, v \in_{k}^{\times}$ ) is developed.