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

Takaaki Kagawa

Displaying similar documents to “The Diophantine Equation X³ = u+v over Real Quadratic Fields”

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 Lebesgue-Nagell equation

Andrzej Dąbrowski (2011)

Colloquium Mathematicae

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We completely solve the Diophantine equations $x ² + 2^{a} q^{b} = y ⁿ$ (for q = 17, 29, 41). We also determine all $C = p ₁^{a ₁} \dots p_{k}^{a_{k}}$ and $C = 2^{a ₀} p ₁^{a ₁} \dots p_{k}^{a_{k}}$ , where $p ₁, . . ., p_{k}$ are fixed primes satisfying certain conditions. The corresponding Diophantine equations x² + C = yⁿ may be studied by the method used by Abu Muriefah et al. (2008) and Luca and Togbé (2009).

Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

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Given a binary recurrence ${u_{n}}_{n \geq 0}$ , we consider the Diophantine equation $u_{n_{1}}^{x_{1}} \dots u_{n_{L}}^{x_{L}} = 1$ with nonnegative integer unknowns $n_{1}, . . ., n_{L}$ , where $n_{i} \neq n_{j}$ for 1 ≤ i < j ≤ L, $m a x | x_{i} | : 1 \leq i \leq L \leq K$ , and K is a fixed parameter. We show that the above equation has only finitely many solutions and the largest one can be explicitly bounded. We demonstrate the strength of our method by completely solving a particular Diophantine equation of the above form.

Finiteness results for Diophantine triples with repdigit values

Attila Bérczes, Florian Luca, István Pink, Volker Ziegler (2016)

Acta Arithmetica

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Let g ≥ 2 be an integer and $_{g} \subset ℕ$ be the set of repdigits in base g. Let $_{g}$ be the set of Diophantine triples with values in $_{g}$ ; that is, $_{g}$ is the set of all triples (a,b,c) ∈ ℕ³ with c < b < a such that ab + 1, ac + 1 and bc + 1 lie in the set $_{g}$ . We prove effective finiteness results for the set $_{g}$ .

On some Diophantine equations involving balancing numbers

Euloge Tchammou, Alain Togbé (2021)

Archivum Mathematicum

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In this paper, we find all the solutions of the Diophantine equation $B_{1}^{p} + 2 B_{2}^{p} + \dots + k B_{k}^{p} = B_{n}^{q}$ in positive integer variables $(k, n)$ , where $B_{i}$ is the $i^{t h}$ balancing number if the exponents $p$ , $q$ are included in the set ${1, 2}$ .

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

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We study the Diophantine equations ${(k!)}^{n} - k^{n} = {(n!)}^{k} - n^{k}$ and ${(k!)}^{n} + k^{n} = {(n!)}^{k} + n^{k},$ where $k$ and $n$ are positive integers. We show that the first one holds if and only if $k = n$ or $(k, n) = (1, 2), (2, 1)$ and that the second one holds if and only if $k = n$ .

On systems of diophantine equations with a large number of solutions

Jerzy Browkin (2010)

Colloquium Mathematicae

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We consider systems of equations of the form $x_{i} + x_{j} = x_{k}$ and $x_{i} \cdot x_{j} = x_{k}$ , which have finitely many integer solutions, proposed by A. Tyszka. For such a system we construct a slightly larger one with much more solutions than the given one.

On the diophantine equations $\prod_{0}^{k} x_{i} - \sum_{0}^{k} x_{i} = n a n d \sum_{0}^{k} 1 / x_{i} = a / n$

Carlo Viola (1973)

Acta Arithmetica

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Solutions of the Diophantine Equation $7 X^{2} + Y^{7} = Z^{2}$ from Recurrence Sequences

Hayder R. Hashim (2020)

Communications in Mathematics

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Consider the system $x^{2} - a y^{2} = b$ , $P (x, y) = z^{2}$ , where $P$ is a given integer polynomial. Historically, the integer solutions of such systems have been investigated by many authors using the congruence arguments and the quadratic reciprocity. In this paper, we use Kedlaya’s procedure and the techniques of using congruence arguments with the quadratic reciprocity to investigate the solutions of the Diophantine equation $7 X^{2} + Y^{7} = Z^{2}$ if $(X, Y) = (L_{n}, F_{n})$ (or $(X, Y) = (F_{n}, L_{n})$ ) where ${F_{n}}$ and ${L_{n}}$ represent the sequences of Fibonacci numbers and Lucas numbers...

The diophantine equation $x^{2} = p^{a} \pm p^{b} + 1$

Florian Luca (2004)

Acta Arithmetica

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On $x^{n} + y^{n} = n! z^{n}$

Susil Kumar Jena (2018)

Communications in Mathematics

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In p. 219 of R.K. Guy’s , 3rd edn., Springer, New York, 2004, we are asked to prove that the Diophantine equation $x^{n} + y^{n} = n! z^{n}$ has no integer solutions with $n \in ℕ_{+}$ and $n > 2$ . But, contrary to this expectation, we show that for $n = 3$ , this equation has infinitely many primitive integer solutions, i.e. the solutions satisfying the condition $gcd (x, y, z) = 1$ .

Diophantine triples with values in binary recurrences

Clemens Fuchs, Florian Luca, Laszlo Szalay (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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In this paper, we study triples $a, b$ and $c$ of distinct positive integers such that $a b + 1, a c + 1$ and $b c + 1$ are all three members of the same binary recurrence sequence.

On the diophantine equation $x^{y} - y^{x} = c^{z}$

Zhongfeng Zhang, Jiagui Luo, Pingzhi Yuan (2012)

Colloquium Mathematicae

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Applying results on linear forms in p-adic logarithms, we prove that if (x,y,z) is a positive integer solution to the equation $x^{y} - y^{x} = c^{z}$ with gcd(x,y) = 1 then (x,y,z) = (2,1,k), (3,2,k), k ≥ 1 if c = 1, and either $(x, y, z) = (c^{k} + 1, 1, k)$ , k ≥ 1 or $2 \leq x < y \leq m a x 1.5 \times 10^{10}, c$ if c ≥ 2.

Corrigendum to the paper “A note on the Diophantine equation $x^{2} + q^{m} = y^{3}$ ” (Acta Arith. 146 (2011), 195-202)

H. L. Zhu (2012)

Acta Arithmetica

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Solutions to xyz = 1 and x + y + z = k in algebraic integers of small degree, II

H. G. Grundman, L. L. Hall-Seelig (2015)

Acta Arithmetica

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Let k ∈ ℤ be such that $|_{k} (ℚ) |$ is finite, where $_{k} : y ² = 1 - 2 k x + k ² x ² - 4 x ³$ . We complete the determination of all solutions to xyz = 1 and x + y + z = k in integers of number fields of degree at most four over ℚ.