V. Ziegler

activity

20102026

most citedOn a variant of Pillai's problem

21 citations · 52 across the 30 of their papers we have counts for

collaborators

37 papers

math.NT2026

On the number of solutions to $S$ -unit equations

Volker Ziegler

Let $K$ be a number field and $S$ a finite set of places including the archimedian ones. Let $(a_{1}, a_{2}, a_{3}) \in K^{*}$ we consider the $S$ -unit equation $a_{1} x + a_{2} y = a_{3}$ with $x,y\in U…

math.NT2026

Complete Resolution Of A Family Of Twisted Thue Equations

Tobias Hilgart, Carina Premstaller, Volker Ziegler

One of the first infinite families of Thue equations, $F_{n} (X) = X^{3} - (n - 1) X^{2} Y - (n + 2) X Y^{2} - Y^{3} = \pm 1$ for $n \in Z$ , was solved by Thomas in 1990. This family is asso…

math.NT2026

On integers that are representable as the sum of two units

Robin Visser, Volker Ziegler

Let $K$ be a number field of degree $D$ with maximal order $O_{K}$ . We show that under certain conditions on $K$ , which in particular are always satisfied if $D$ is odd or…

math.NT2023

Twisted Thue equations with multiple exponents in fixed number fields

Tobias Hilgart, Volker Ziegler

Let $K$ be a number field of degree $d \geq 3$ and fix $s$ multiplicatively independent algebraic integers $γ_{1}, \dots, γ_{s} \in K^{*}$ that fulfil some technical requirements, which c…

math.NT2022★ 2 cited

On the Diophantine equation $U_{n} - b^{m} = c$

Sebastian Heintze, Robert F. Tichy, Ingrid Vukusic +1

Let $(U_{n})_{n \in N}$ be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable c…

math.NT2021★ 2 cited

On a family of cubic Thue Equations involving Fibonacci and Lucas numbers

Tobias Hilgart, Ingrid Vukusic, Volker Ziegler

Let $F_{n}$ denote the $n$ -th Fibonacci number and $L_{n}$ the $n$ -th Lucas number. We completely solve the family of cubic Thue equations $(X - F_{n} Y) (X - L_{n} Y) X - Y^{3} = \pm 1$ and show tha…

math.NT2021★ 1 cited

On sums of Fibonacci numbers with few binary digits

Ingrid Vukusic, Volker Ziegler

In this paper we completely solve the Diophantine equation $F_{n} + F_{m} = 2^{a_{1}} + 2^{a_{2}} + 2^{a_{3}} + 2^{a_{4}} + 2^{a_{5}}$ , where $F_{k}$ denotes the $k$ -th Fibonacci number. In addition to comple…

math.NT2021

On a family of unit equations over simplest cubic fields

Ingrid Vukusic, Volker Ziegler

Let $a \in Z$ and $ρ$ be a root of $f_{a} (x) = x^{3} - a x^{2} - (a + 3) x - 1$ , then the number field $K_{a} = Q (ρ)$ is called a simplest cubic field. In this paper we consider the fam…

math.NT2021

Sums of Fibonacci numbers that are perfect powers

Volker Ziegler

Let us denote by $F_{n}$ the $n$ -th Fibonacci number. In this paper we show that for a fixed integer $y$ there exists at most one integer exponent $a > 0$ such that the Diophantine equ…

math.NT2021

Consecutive tuples of multiplicatively dependent integers

Ingrid Vukusic, Volker Ziegler

This paper is concerned with the existence of consecutive pairs and consecutive triples of multiplicatively dependent integers. A theorem by LeVeque on Pillai's equation implies th…

math.NT2020

On a variant of Pillai's problem with transcendental numbers

Robert Tichy, Ingrid Vukusic, Daodao Yang +1

In this paper, we study the asymptotic behaviour of the number of solutions $(m, n) \in N^{2}$ to the inequality $∣ α^{n} - β^{m} ∣ \leq x$ when $x$ tends to infinity. Here $α,…

math.NT2020

Finding all $S$ -Diophantine quadruples for a fixed set of primes $S$

Volker Ziegler

Given a finite set of primes $S$ and a $m$ -tuple $(a_{1}, \dots, a_{m})$ of positive, distinct integers we call the $m$ -tuple $S$ -Diophantine, if for each $1 \leq i < j \leq m$ the quantit…

math.NT2020

Integers representable as differences of linear recurrence sequences

Robert Tichy, Ingrid Vukusic, Daodao Yang +1

Let ${U_{n}}_{n \geq 0}$ and ${V_{m}}_{m \geq 0}$ be two linear recurrence sequences. We establish an asymptotic formula for the number of integers $c$ in the range $[- x, x]$ which…

math.NT2020

Integers representable as differences of linear recurrence sequences

Daodao Yang

Let ${U_{n}}_{n ⩾ 0}$ and ${G_{m}}_{m ⩾ 0}$ be two linear recurrence sequences defined over the integers. We establish an asymptotic formula for the number of integ…

math.NT2020★ 2 cited

Irrationality of growth constants associated with polynomial recursions

Stephan Wagner, Volker Ziegler

We consider integer sequences that satisfy a recursion of the form $x_{n + 1} = P (x_{n})$ for some polynomial $P$ of degree $d > 1$ . If such a sequence tends to infinity, then it satis…

math.NT2018★ 1 cited

On the existence of $S$ -Diophantine quadruples

Volker Ziegler

Let $S$ be a set of primes. We call an $m$ -tuple $(a_{1}, \dots, a_{m})$ of distinct, positive integers $S$ -Diophantine, if for all $i \neq = j$ the integers $s_{i, j} := a_{i} a_{j} + 1$ have only…

math.NT2018

Effective results for linear Equations in Members of two Recurrence Sequences

Volker Ziegler

Let $(U_{n})_{n = 0}^{\infty}$ and $(V_{m})_{m = 0}^{\infty}$ be two linear recurrence sequences. For fixed positive integers $k$ and $ℓ$ , fixed $k$ -tuple $(a_{1}, \dots, a_{k}) \in Z^{k}$ …

math.NT2017★ 1 cited

On Diophantine equations involving sums of Fibonacci numbers and powers of $2$

Kwok Chi Chim, Volker Ziegler

In this paper, we completely solve the Diophantine equations $F_{n_{1}} + F_{n_{2}} = 2^{a_{1}} + 2^{a_{2}} + 2^{a_{3}}$ and $F_{m_{1}} + F_{m_{2}} + F_{m_{3}} = 2^{t_{1}} + 2^{t_{2}}$ , where $F_{k}$ d…

math.NT2017

On arithmetic progressions in Lucas sequences

Lajos Hajdu, Márton Szikszai, Volker Ziegler

In this paper, we consider arithmetic progressions contained in Lucas sequences of first and second kind. We prove that for almost all sequences, there are only finitely many and t…

math.NT2016

On a variant of Pillai's problem II

Kwok Chi Chim, István Pink, Volker Ziegler

In this paper, we show that there are only finitely many $c$ such that the equation $U_{n} - V_{m} = c$ has at least two distinct solutions $(n, m)$ , where ${U_{n}}_{n \geq 0}$ and $\{V_…

math.NT2016★ 2 cited

There is no Diophantine quintuple

Bo He, Alain Togbè, Volker Ziegler

A set of $m$ positive integers ${a_{1}, a_{2}, \dots, a_{m}}$ is called a Diophantine $m$ -tuple if $a_{i} a_{j} + 1$ is a perfect square for all $1 \leq i < j \leq m$ . In 2004 Dujella prove…

math.NT2016★ 1 cited

Effective Resolution of Diophantine equations of the form $u_{n} + u_{m} = w p_{1}^{z_{1}} \dots p_{s}^{z_{s}}$

István Pink, Volker Ziegler

Let $u_{n}$ be a fixed non-degenerate binary recurrence sequence with positive discriminant, $w$ a fixed non-zero integer and $p_{1}, p_{2}, \dots, p_{s}$ fixed, distinct prime numbers. In th…

math.NT2016★ 21 cited

On a variant of Pillai's problem

Kwok Chi Chim, István Pink, Volker Ziegler

In this paper, we find all integers $c$ having at least two representations as a difference between a Fibonacci number and a Tribonacci number.

math.NT2016

Non-Minimality of the Width- $w$ Non-adjacent Form in Conjunction with Trace One $τ$ -adic Digit Expansions and Koblitz Curves in Characteristic Two

Daniel Krenn, Volker Ziegler

This article deals with redundant digit expansions with an imaginary quadratic algebraic integer with trace $\pm 1$ as base and a minimal norm representatives digit set. For $w\geq…

On the number of solutions to S-unit equations

Complete Resolution Of A Family Of Twisted Thue Equations

On integers that are representable as the sum of two units

Twisted Thue equations with multiple exponents in fixed number fields

On the Diophantine equation Un​−bm=c

On a family of cubic Thue Equations involving Fibonacci and Lucas numbers

On sums of Fibonacci numbers with few binary digits

On a family of unit equations over simplest cubic fields

Sums of Fibonacci numbers that are perfect powers

Consecutive tuples of multiplicatively dependent integers

On a variant of Pillai's problem with transcendental numbers

Finding all S-Diophantine quadruples for a fixed set of primes S

Integers representable as differences of linear recurrence sequences

Integers representable as differences of linear recurrence sequences

Irrationality of growth constants associated with polynomial recursions

On the existence of S-Diophantine quadruples

Effective results for linear Equations in Members of two Recurrence Sequences

On Diophantine equations involving sums of Fibonacci numbers and powers of 2

On arithmetic progressions in Lucas sequences

On a variant of Pillai's problem II

There is no Diophantine quintuple

Effective Resolution of Diophantine equations of the form un​+um​=wp1z1​​⋯pszs​​

On a variant of Pillai's problem

Non-Minimality of the Width-w Non-adjacent Form in Conjunction with Trace One τ-adic Digit Expansions and Koblitz Curves in Characteristic Two

On the number of solutions to $S$ -unit equations

On the Diophantine equation $U_{n} - b^{m} = c$

Finding all $S$ -Diophantine quadruples for a fixed set of primes $S$

On the existence of $S$ -Diophantine quadruples

On Diophantine equations involving sums of Fibonacci numbers and powers of $2$

Effective Resolution of Diophantine equations of the form $u_{n} + u_{m} = w p_{1}^{z_{1}} \dots p_{s}^{z_{s}}$

Non-Minimality of the Width- $w$ Non-adjacent Form in Conjunction with Trace One $τ$ -adic Digit Expansions and Koblitz Curves in Characteristic Two