arXiv subjects
Publications and source records attributed to 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 complex linear forms in logarithms and the Baker-Davenport reduction method, we use $p$-adic versions of both tools.
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 equation $F_n+F_m=y^a$ has a solution $(n,m,a)$ in positive integers satisfying $n>m>0$, unless $y=2,3,4,6$ or $10$.
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 that the only consecutive pairs of multiplicatively dependent integers larger than 1 are $(2,8)$ and $(3,9)$. For triples, we prove the following theorem: If $a\notin \{2,8\}$ is a fixed integer larger than 1, then there are only finitely many triples $(a,b,c)$ of pairwise distinct integers larger than 1 such that $(a,b,c)$, $(a+1,b+1,c+1)$ and $(a+2,b+2,c+2)$ are each multiplicatively dependent. Moreover, these triples can be determined effectively.
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 quantity $a_ia_j+1$ has prime divisors coming only from the set $S$. For a given set $S$ we give a practical algorithm to find all $S$-Diophantine quadruples, provided that $|S|=3$.
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 satisfies an asymptotic formula of the form $x_n \sim A α^{d^n}$, but little can be said about the constant $α$. In this paper, we show that $α$ is always irrational or an integer. In fact, we prove a stronger statement: if a sequence $G_n$ satisfies an asymptotic formula of the form $G_n = A α^n + B + O(α^{-εn})$, where $A,B$ are algebraic and $α> 1$, and the sequence contains infinitely many integers, then $α$ is irrational or an integer.
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 can be represented as differences $ U_n - V_m$. In particular, the density of such integers is $0$.
Developing products, services and vertical applications for the future digitized society in the 6G era requires a multidisciplinary approach and a re-definition of how we create, deliver and consume network resources, data and services for both communications and sensing purposes. This development will change and disrupt the traditional business models and ecosystem roles of digital service providers, as well as open the market for key stakeholders in the 6G era like digital service operators, cloud operators and resource brokers. White paper discusses unprecedented opportunities of enabling and empowering multiple stakeholders to have a more active participation in the future 6G ecosystem via novel sustainable open ecosystemic business models with flexible integration of long tail services with tailored performance attributes. This research adopts a qualitative scenario planning method and portrays three scenario themes resulting in a total of 12 scenarios for the futures of the 6G business. By focusing on key trends, their interactions, and irreducible uncertainties, scenario building generates perspectives for the futures within which alternative 6G business strategies were developed and assessed for a traditional incumbent mobile network operator and a novel 6G digital service provider stemming from redefined sustainable economics. Value-capture in the 6G era requires understanding the dynamics of platforms and ecosystems. Results indicate that, to reach some of the preferred futures, we should pay attention to the privacy and security issues related to business and regulation needs; public/governmental, corporate, community and user(s) perspectives to and aims of governance; ecosystem configuration related to users, decentralized business models and platforms; user empowerment; and the role of location-specificity of services.
The commercial launch of 6G communications systems and United Nations Sustainable Development Goals, UN SDGs, are both targeted for 2030. 6G communications is expected to boost global growth and productivity, create new business models and transform many aspects of society. The UN SDGs are a way of framing opportunities and challenges of a desirable future world and cover topics as broad as ending poverty, gender equality, climate change and smart cities. The relationship between these potentially mutually reinforcing forces is currently under-defined. Building on the vision for 6G, a review of megatrends, on-going activities on the relation of mobile communications to the UN SDGs and existing indicators, a novel linkage between 6G and the UN SDGs is proposed via indicators. The white paper has also launched the work of deriving new 6G related indicators to guide the research of 6G systems. The novel linkage is built on the envisaged three-fold role of 6G as a provider of services to help steer and support communities and countries towards reaching the UN SDGs, as an enabler of measuring tool for data collection to help reporting of indicators with hyperlocal granularity, and as a reinforcer of new ecosystems based on 6G technology enablers and 6G network of networks to be developed in line with the UN SDGs that incorporates future mobile communication technologies available in 2030. Related challenges are also identified. An action plan is presented along with prioritized focus areas within the mobile communication sector technology and industry evolution to best support the achievement of the UN SDGs.
Let $S$ be a set of primes. We call an $m$-tuple $(a_1,\ldots,a_m)$ of distinct, positive integers $S$-Diophantine, if for all $i\neq j$ the integers $s_{i,j}:=a_ia_j+1$ have only prime divisors coming from the set $S$, i.e. if all $s_{i,j}$ are $S$-units. In this paper, we show that no $S$-Diophantine quadruple (i.e.~$m=4$) exists if $S=\{3,q\}$. Furthermore we show that for all pairs of primes $(p,q)$ with $p<q$ and $p\equiv 3\mod 4$ no $\{p,q\}$-Diophantine quadruples exist, provided that $(p,q)$ is not a Wieferich prime pair.
Let $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ be two linear recurrence sequences. For fixed positive integers $k$ and $\ell$, fixed $k$-tuple $(a_1,\dots,a_k)\in \mathbb{Z}^k$ and fixed $\ell$-tuple $(b_1,\dots,b_\ell)\in \mathbb{Z}^\ell$ we consider the linear equation $$a_1U_{n_1}+\cdots +a_k U_{n_k}=b_1V_{m_1}+\cdots + b_\ell V_{m_\ell}$$ in the unknown non-negative integers $n_1,\dots,n_k$ and $m_1,\dots,m_\ell$. Under the assumption that the linear recurrences $(U_n)_{n=0}^\infty$ and $(V_m)_{m=0}^\infty$ have dominant roots and under the assumption of further mild restrictions we show that this equation has only finitely many solutions which can be found effectively.
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 $1 \le i < j \le m$. In 2004 Dujella proved that there is no Diophantine sextuple and that there are at most finitely many Diophantine quintuples. In particular, a folklore conjecture concerning Diophantine $m$-tuples states that no Diophantine quintuple exists at all. In this paper we prove this conjecture.
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 their number can be effectively bounded. We also show that there are only a few sequences which contain infinitely many and one can explicitly list both the sequences and the progressions in them. A more precise statement is given for sequences with dominant root.
We prove an asymptotic formula for the number of S-integers in a number field K that can be represented by a sum of n S-integers of bounded norm.
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 $F_k$ denotes the $k$-th Fibonacci number. In particular, we prove that $\max \{n_1, n_2, a_1, a_2, a_3 \}\leq 18$ and $\max \{ m_1, m_2, m_3, t_1, t_2 \}\leq 16$.
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 $\{V_m\}_{m\geq 0}$ are given linear recurrence sequences.
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 $w\geq 2$ it is shown that the width-$w$ non-adjacent form is not an optimal expansion, meaning that it does not minimize the (Hamming-)weight among all possible expansions with the same digit set. One main part of the proof uses tools from Diophantine analysis, namely the theory of linear forms in logarithms and the Baker--Davenport reduction method.
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 this paper we consider the Diophantine equation $u_n+u_m=w p_1^{z_1} \cdots p_s^{z_s}$ and prove under mild technical restrictions effective finiteness results. In particular we give explicit upper bounds for $n,m$ and $z_1, \dots, z_s$. Furthermore, we provide a rather efficient algorithm to solve Diophantine equations of the described type and we demonstrate our method by an example.
In this paper, we find all integers $c$ having at least two representations as a difference between a Fibonacci number and a Tribonacci number.