On systems of diophantine equations with a large number of solutions

Jerzy Browkin

Displaying similar documents to “On systems of diophantine equations with a large number of solutions”

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 $_{g}$ .

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 $k = n$ .

A note on the article by F. Luca “On the system of Diophantine equations $a ² + b ² = {(m ² + 1)}^{r}$ and $a^{x} + b^{y} = {(m ² + 1)}^{z}$ ” (Acta Arith. 153 (2012), 373-392)

Takafumi Miyazaki (2014)

Acta Arithmetica

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Let r,m be positive integers with r > 1, m even, and A,B be integers satisfying $A + B \sqrt (- 1) = {(m + \sqrt (- 1))}^{r}$ . We prove that the Diophantine equation ${| A |}^{x} + {| B |}^{y} = {(m ² + 1)}^{z}$ has no positive integer solutions in (x,y,z) other than (x,y,z) = (2,2,r), whenever $r > 10^{74}$ or $m > 10^{34}$ . Our result is an explicit refinement of a theorem due to F. Luca.

Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva (2013)

Journal de Théorie des Nombres de Bordeaux

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Let $F_{n}$ be the $n$ -th Fibonacci number. Put $ϕ = \frac{1 + \sqrt{5}}{2}$ . We prove that the following inequalities hold for any real $α$ : 1) ${inf}_{n \in ℕ} | | F_{n} α | | \leq \frac{ϕ - 1}{ϕ + 2}$ , 2) ${lim inf}_{n \to \infty} | | F_{n} α | | \leq \frac{1}{5}$ , 3) ${lim inf}_{n \to \infty} | | ϕ^{n} α | | \leq \frac{1}{5}$ . These results are the best possible.

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.

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.

The number of solutions to the generalized Pillai equation $\pm r a^{x} \pm s b^{y} = c$ .

Reese Scott, Robert Styer (2013)

Journal de Théorie des Nombres de Bordeaux

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We consider $N$ , the number of solutions $(x, y, u, v)$ to the equation ${(- 1)}^{u} r a^{x} + {(- 1)}^{v} s b^{y} = c$ in nonnegative integers $x, y$ and integers $u, v \in {0, 1}$ , for given integers $a > 1$ , $b > 1$ , $c > 0$ , $r > 0$ and $s > 0$ . When $gcd (r a, s b) = 1$ , we show that $N \leq 3$ except for a finite number of cases all of which satisfy $max (a, b, r, s, x, y) < 2 \cdot 10^{15}$ for each solution; when $gcd (a, b) > 1$ , we show that $N \leq 3$ except for three infinite families of exceptional cases. We find several different ways to generate an infinite number of cases giving $N = 3$ solutions.

A note on the weighted Khintchine-Groshev Theorem

Mumtaz Hussain, Tatiana Yusupova (2014)

Journal de Théorie des Nombres de Bordeaux

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Let $W (m, n; \underset{̲}{ψ})$ denote the set of $ψ_{1}, ..., ψ_{n}$ –approximable points in $ℝ^{m n}$ . The classical Khintchine–Groshev theorem assumes a monotonicity condition on the approximating functions $\underset{̲}{ψ}$ . Removing monotonicity from the Khintchine–Groshev theorem is attributed to different authors for different cases of $m$ and $n$ . It can not be removed for $m = n = 1$ as Duffin–Schaeffer provided the counter example. We deal with the only remaining case $m = 2$ and thereby remove all unnecessary conditions from the Khintchine–Groshev theorem. ...

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

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The Littlewood conjecture in Diophantine approximation claims that $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot ∥ q β ∥ = 0$ holds for all real numbers $α$ and $β$ , where $∥ \cdot ∥$ denotes the distance to the nearest integer. Its $p$ -adic analogue, formulated by de Mathan and Teulié in 2004, asserts that $inf_{q \geq 1} q \cdot ∥ q α ∥ \cdot {| q |}_{p} = 0$ holds for every real number $α$ and every prime number $p$ , where ${| \cdot |}_{p}$ denotes the $p$ -adic absolute value normalized by ${| p |}_{p} = p^{- 1}$ . We survey the known results on these conjectures and highlight recent developments. ...

Rigidity of critical circle mappings I

Edson de Faria, Welington de Melo (1999)

Journal of the European Mathematical Society

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We prove that two $C^{3}$ critical circle maps with the same rotation number in a special set $𝔸$ are $C^{1 + α}$ conjugate for some $α > 0$ provided their successive renormalizations converge together at an exponential rate in the $C^{0}$ sense. The set $𝔸$ has full Lebesgue measure and contains all rotation numbers of bounded type. By contrast, we also give examples of $C^{\infty}$ critical circle maps with the same rotation number that are not $C^{1 + β}$ conjugate for any $β > 0$ . The class of rotation numbers for which such examples exist...

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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Inhomogeneous Diophantine approximation with general error functions

Lingmin Liao, Michał Rams (2013)

Acta Arithmetica

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Let α be an irrational and φ: ℕ → ℝ⁺ be a function decreasing to zero. Let $ω (α) : = s u p θ \geq 1 : l i m i n f_{n \to \infty} n^{θ} | | n α | = 0$ $. F o r a n y α w i t h a g i v e n ω (α), w e g i v e s o m e s h a r p e s t i m a t e s f o r t h e H a u s d o r f f d i m e n s i o n o f t h e s e t$ $E_{φ} (α)$ := y ∈ ℝ: ||nα -y|| < φ(n) for infinitely many n, where ||·|| denotes the distance to the nearest integer.