Random Thue and Fermat equations

Rainer Dietmann; Oscar Marmon

Displaying similar documents to “Random Thue and Fermat equations”

On the Brocard-Ramanujan problem and generalizations

Andrzej Dąbrowski (2012)

Colloquium Mathematicae

Similarity:

Let $p_{i}$ denote the ith prime. We conjecture that there are precisely 28 solutions to the equation $n ² - 1 = p ₁^{α ₁} \dots p_{k}^{α_{k}}$ in positive integers n and α₁,..., $α_{k}$ . This conjecture implies an explicit description of the set of solutions to the Brocard-Ramanujan equation. We also propose another variant of the Brocard-Ramanujan problem: describe the set of solutions in non-negative integers of the equation n! + A = x₁²+x₂²+x₃² (A fixed).

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

Similarity:

Multiplicative relations on binary recurrences

Florian Luca, Volker Ziegler (2013)

Acta Arithmetica

Similarity:

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.

Diophantine triples with values in binary recurrences

Clemens Fuchs, Florian Luca, Laszlo Szalay (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

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.

Finiteness results for Diophantine triples with repdigit values

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

Acta Arithmetica

Similarity:

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 the Lebesgue-Nagell equation

Andrzej Dąbrowski (2011)

Colloquium Mathematicae

Similarity:

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).

Metric Diophantine approximation on the middle-third Cantor set

Yann Bugeaud, Arnaud Durand (2016)

Journal of the European Mathematical Society

Similarity:

Let $μ \geq 2$ be a real number and let $ℳ (μ)$ denote the set of real numbers approximable at order at least $μ$ by rational numbers. More than eighty years ago, Jarník and, independently, Besicovitch established that the Hausdorff dimension of $ℳ (μ)$ is equal to $2 / μ$ . We investigate the size of the intersection of $ℳ (μ)$ with Ahlfors regular compact subsets of the interval $[0, 1]$ . In particular, we propose a conjecture for the exact value of the dimension of $ℳ (μ)$ intersected with the middle-third Cantor set and give several...

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

H. G. Grundman, L. L. Hall-Seelig (2014)

Acta Arithmetica

Similarity:

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

On systems of diophantine equations with a large number of solutions

Jerzy Browkin (2010)

Colloquium Mathematicae

Similarity:

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.

Diophantine equations involving factorials

Horst Alzer, Florian Luca (2017)

Mathematica Bohemica

Similarity:

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 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

Similarity:

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

Similarity:

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.

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

Florian Luca (2004)

Acta Arithmetica

Similarity:

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

Similarity:

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

Zhongfeng Zhang, Jiagui Luo, Pingzhi Yuan (2012)

Colloquium Mathematicae

Similarity:

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 generalized Fermat equations of signature (p,p,3)

Karolina Krawciów (2011)

Colloquium Mathematicae

Similarity:

This paper focuses on the Diophantine equation $x ⁿ + p^{α} y ⁿ = M z ³$ , with fixed α, p, and M. We prove that, under certain conditions on M, this equation has no non-trivial integer solutions if $n \geq ℱ (M, p^{α})$ , where $ℱ (M, p^{α})$ is an effective constant. This generalizes Theorem 1.4 of the paper by Bennett, Vatsal and Yazdani [Compos. Math. 140 (2004), 1399-1416].

Further remarks on Diophantine quintuples

Mihai Cipu (2015)

Acta Arithmetica

Similarity:

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)...

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

Similarity:

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.

Around the Littlewood conjecture in Diophantine approximation

Yann Bugeaud (2014)

Publications mathématiques de Besançon

Similarity:

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. ...