Diophantine approximations with Fibonacci numbers

Victoria Zhuravleva

Displaying similar documents to “Diophantine approximations with Fibonacci numbers”

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 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 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. ...

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.

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 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 operators from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ or to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$

Christian Samuel (2010)

Colloquium Mathematicae

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We show that every operator from $ℓ_{s}$ to $ℓ_{p} \otimes ̂ ℓ_{q}$ is compact when 1 ≤ p,q < s and that every operator from $ℓ_{s}$ to $ℓ_{p} \hat{\otimes ̂} ℓ_{q}$ is compact when 1/p + 1/q > 1 + 1/s.

Integers not of the form $c (2^{a} + 2^{b}) + p^{α}$

Zhi-Wei Sun, Mao-Hua Le (2001)

Acta Arithmetica

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On the equation $ϕ (x + m) = ϕ (x) + m$

A. Makowski (1964)

Matematički Vesnik

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The existence of $P (α)$ -points of N* for $ℵ_{0} α < 𝔠$

A. Szymański (1977)

Colloquium Mathematicae

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On C $^{*}$ -spaces

P. Srivastava, K. K. Azad (1981)

Matematički Vesnik

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$ℵ_{1}$ -incompactness of Z

Ralph McKenzie (1971)

Colloquium Mathematicae

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On a characterization of $L^{p}$ -norm

Janusz Matkowski (1989)

Annales Polonici Mathematici

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On the $p^{λ}$ problem

Stephan Baier (2004)

Acta Arithmetica

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Геометрическая теория состояний, граница фон Неймана, двойственность $C^{*}$ -алгерб

А.М. Вершик (1972)

Zapiski naucnych seminarov Leningradskogo

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An application of the method wherein the $p h i$ - and $c h i$ - methods are combined for the determination of the grating constant. [II.]

Swami Jnanananda (1936)

Časopis pro pěstování matematiky a fysiky

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On the proximate order of $f^{(s)} (z) * g^{(s)} (z)$ and ${[f (z) * g (z)]}^{(s)}$

M. K. Sen (1971)

Annales Polonici Mathematici

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On spirallike functions of order $α$ and type $β$ with fixed second coefficient

M. K. Aouf (1988)

Matematički Vesnik

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The general methods of finding the sum $S_{n}$ for all kinds of series, II

K. Orlov (1981)

Matematički Vesnik

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Approximation properties of β-expansions

Simon Baker (2015)

Acta Arithmetica

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Let β ∈ (1,2) and x ∈ [0,1/(β-1)]. We call a sequence ${(ϵ_{i})}_{i = 1}^{\infty} \in {0, 1}^{ℕ}$ a β-expansion for x if $x = \sum_{i = 1}^{\infty} ϵ_{i} β^{- i}$ . We call a finite sequence ${(ϵ_{i})}_{i = 1}^{n} \in {0, 1}^{n}$ an n-prefix for x if it can be extended to form a β-expansion of x. In this paper we study how good an approximation is provided by the set of n-prefixes. Given $Ψ : ℕ \to ℝ_{\geq 0}$ , we introduce the following subset of ℝ: $W_{β} (Ψ) : = ⋂_{m = 1}^{\infty} ⋃_{n = m}^{\infty} ⋃_{{(ϵ_{i})}_{i = 1}^{n} \in {0, 1}^{n}} [\sum_{i = 1}^{n} (ϵ_{i}) / (β^{i}), \sum_{i = 1}^{n} (ϵ_{i}) / (β^{i}) + Ψ (n)]$ In other words, $W_{β} (Ψ)$ is the set of x ∈ ℝ for which there exist infinitely many solutions to the inequalities $0 \leq x - \sum_{i = 1}^{n} (ϵ_{i}) / (β^{i}) \leq Ψ (n)$ . When $\sum_{n = 1}^{\infty} 2^{n} Ψ (n) < \infty$ , the Borel-Cantelli lemma tells us that the Lebesgue measure...