Attila Pethő
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View article: Decomposable forms generated by linear recurrences
Decomposable forms generated by linear recurrences Open
Consider $k\ge 2$ distinct, linearly independent, homogeneous linear recurrences of order $k$ satisfying the same recurrence relation. We prove that the recurrences are related to a decomposable form of degree $k$, and there is a very broa…
View article: Upper bound on the solution to F(2k)n = +F(2k)m with negative subscripts
Upper bound on the solution to F(2k)n = +F(2k)m with negative subscripts Open
In this paper, we provide an explicit upper bound on the absolute value of the solutions n < m < 0 to the Diophantine equation F(k)n = ±F(k)m, assuming k is even. Here {F(k)n}n ∈ Z denotes the k-generalized Fibonacci sequence. The upper bo…
View article: Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters
Asymptotics of $D(q)$-pairs and triples via $L$-functions of Dirichlet charaters Open
Let $q$ be an integer. A $D(q)$-$m$-tuple is a set of $m$ distinct positive integers ${a_1, a_2, . . . , a_m}$ such that $a_ia_j + q$ is a perfect square for all $1 \leq i < j \leq m$. By counting integer solutions $x \in [1, b]$ of congru…
View article: Common values of a class of linear recurrence
Common values of a class of linear recurrence Open
Let $(a_n), (b_n)$ be linear recursive sequences of integers with characteristic polynomials $A(X),B(X)\in \mathbb{Z}[X]$ respectively. Assume that $A(X)$ has a dominating and simple real root $α$, while $B(X)$ has a pair of conjugate comp…
View article: On $k$-generalized Fibonacci numbers with negative indices
On $k$-generalized Fibonacci numbers with negative indices Open
In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even then the abs…
View article: On the $k$-generalized Fibonacci numbers with negative indices
On the $k$-generalized Fibonacci numbers with negative indices Open
In these notes we study the $k$-generalized Fibonacci sequences - $(F_n^{(k)})_{n\in \Z}$ - with positive and negative indices. Denote $T_k(x)$ its characteristic polynomial. Our most interesting finding is that if $k$ is even then the abs…
View article: The Finiteness Property for Shift Radix Systems With General Parameters
The Finiteness Property for Shift Radix Systems With General Parameters Open
See the abstract in the attached pdf.
View article: Number systems over general orders
Number systems over general orders Open
Let $\mathcal{O}$ be an order, that is a commutative ring with $1$ whose additive structure is a free $\mathbb{Z}$-module of finite rank. A generalized number system (GNS for short) over $\mathcal{O}$ is a pair $(p,\mathcal{D} )$ where $p\…
View article: Cross-platform Identity-based Cryptography using WebAssembly
Cross-platform Identity-based Cryptography using WebAssembly Open
The explosive spread of the devices connected to the Internet has increased the need for efficient and portable cryptographic routines.Despite this fact, truly platform-independent implementations are still hard to find.In this paper, an I…
View article: Shift radix systems with general parameters
Shift radix systems with general parameters Open
There are two dimensional expanding SRS, which have some periodic orbits. The aim of the present note is to describe as good as possible such unusual points. We give all regions, to which points belong obvious cycles, like $(1), (-1), (1,-…
View article: The finiteness property for shift radix systems with general parameters
The finiteness property for shift radix systems with general parameters Open
There are two-dimensional expanding shift radix systems (SRS) which have some periodic orbits. The aim of the present paper is to describe such unusual points as well as possible. We give all regions that contain parameters the correspondi…
View article: On multidimensional Diophantine approximation of algebraic numbers
On multidimensional Diophantine approximation of algebraic numbers Open
In this article we develop algorithms for solving the dual problems of approximating linear forms and of simultaneous approximation in number fields F. Using earlier ideas for computing independent units by Buchmann, Pethő and later Pohst …
View article: Canonical number systems over imaginary quadratic Euclidean domains
Canonical number systems over imaginary quadratic Euclidean domains Open
We investigate canonical number systems over imaginary quadratic Euclidean domains. We define a canonical digit set in a uniform way. Linear ECNS polynomials are characterized completely. We prove that for every degree there are infinitely…
View article: On nearly linear recurrence sequences
On nearly linear recurrence sequences Open
A nearly linear recurrence sequence (nlrs) is a complex sequence $(a_n)$ with the property that there exist complex numbers $A_0$,$\ldots$, $A_{d-1}$ such that the sequence $\big(a_{n+d}+A_{d-1}a_{n+d-1}+\cdots +A_0a_n\big)_{n=0}^{\infty}$…