Artin's primitive root conjecture -a survey -

This is an expanded version of a write-up of a talk given in the fall of 2000 in Oberwolfach. A large part of it is intended to be understandable by non-number theorists with a mathematical background. The talk covered some of the history, results and ideas connected with Artin's celebrated primitive root conjecture dating from 1927. In the update several new results established after 2000 are also discussed.Comment: 87 pages, 512 references, to appear in Integer

Semiconjugate Factorizations of Higher Order Linear Difference Equations in Rings

We study linear difference equations with variable coefficients in a ring using a new nonlinear method. In a ring with identity, if the homogeneous part of the linear equation has a solution in the unit group of the ring (i.e., a unitary solution) then we show that the equation decomposes into two linear equations of lower orders. This decomposition, known as a semiconjugate factorization in the nonlinear theory, generalizes the classical operator factorization in the linear context. Sequences of ratios of consecutive terms of a unitary solution are used to obtain the semiconjugate factorization. Such sequences, known as eigensequences are well-suited to variable coefficients; for instance, they provide a natural context for the expression of the classical Poincar\'{e}-Perron Theorem. We discuss some applications to linear difference equations with periodic coefficients and also derive formulas for the general solutions of linear functional recurrences satisfied by the classical special functions such as the modified Bessel and Chebyshev.Comment: Application of nonlinear semiconjugate factorization theory to linear difference equations with variable coefficients in rings; 29 pages, containing the main theory and more than 8 examples worked out in detai

NumGfun: a Package for Numerical and Analytic Computation with D-finite Functions

This article describes the implementation in the software package NumGfun of classical algorithms that operate on solutions of linear differential equations or recurrence relations with polynomial coefficients, including what seems to be the first general implementation of the fast high-precision numerical evaluation algorithms of Chudnovsky & Chudnovsky. In some cases, our descriptions contain improvements over existing algorithms. We also provide references to relevant ideas not currently used in NumGfun

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)

Almost all primes have a multiple of small Hamming weight

Recent results of Bourgain and Shparlinski imply that for almost all primes $p$ there is a multiple $mp$ that can be written in binary as $mp= 1+2^{m_1}+ \cdots +2^{m_k}, \quad 1\leq m_1 < \cdots < m_k,$ with $k=66$ or $k=16$, respectively. We show that $k=6$ (corresponding to Hamming weight $7$) suffices. We also prove there are infinitely many primes $p$ with a multiplicative subgroup $A=\subset \mathbb{F}_p^*$, for some $g \in \{2,3,5\}$, of size $|A|\gg p/(\log p)^3$, where the sum-product set $A\cdot A+ A\cdot A$ does not cover $\mathbb{F}_p$ completely