Linear Recurrence Sequences Research Articles

Greatest common divisors and Vojta’s conjecture for blowups of algebraic tori

We give results and inequalities bounding the greatest common divisor of multivariable polynomials evaluated at S-unit arguments, generalizing to an arbitrary number of variables results of Bugeaud–Corvaja–Zannier, Hernandez–Luca, and Corvaja–Zannier. In closely related results, and in line with observations of Silverman, we prove special cases of Vojta’s conjecture for blowups of toric varieties. As an application, we classify when terms from simple linear recurrence sequences can have a large greatest common divisor (in an appropriate sense). The primary tool used in the proofs is Schmidt’s Subspace Theorem from Diophantine approximation.

Decomposable polynomials in second order linear recurrence sequences

We study elements of second order linear recurrence sequences (G_n)_{n= 0}^{infty } of polynomials in {{mathbb {C}}}[x] which are decomposable, i.e. representable as G_n=gcirc h for some g, hin {{mathbb {C}}}[x] satisfying deg g,deg h>1. Under certain assumptions, and provided that h is not of particular type, we show that deg g may be bounded by a constant independent of n, depending only on the sequence.

Effective results on the Skolem Problem for linear recurrence sequences

In this paper, given a simple linear recurrence sequence of algebraic numbers, which has either a dominant characteristic root or exactly two characteristic roots of maximal modulus, we give some explicit lower bounds for the index beyond which every term of the sequence is non-zero. It turns out that this case covers almost all such sequences whose coefficients are rational numbers.

Diophantine triples in linear recurrences of Pisot type

The study of Diophantine triples taking values in linear recurrence sequences is a variant of a problem going back to Diophantus of Alexandria which has been studied quite a lot in the past. The main questions are, as usual, about existence or finiteness of Diophantine triples in such sequences. Whilst the case of binary recurrence sequences is almost completely solved, not much was known about recurrence sequences of larger order, except for very specialised generalisations of the Fibonacci sequence. Now, we will prove that any linear recurrence sequence with the Pisot property contains only finitely many Diophantine triples, whenever the order is large and a few more not very restrictive conditions are met.

Construction of Uniformly Distributed Linear Recurring Sequences Modulo Powers of 2

AbstractThe aim of the present paper is to provide the background to construct linear recurring sequences with uniform distribution modulo 2s. The theory is developed and an algorithm based on the achieved results is given. The constructed sequences may have arbitrary large period length depending only on the computational power of the used machines.

On the Some Properties of Circulant Matrices with Third Order Linear Recurrent Sequences

In this paper, firstly, we give the some fundamental properties of Van Der Laan numbers. After, we define the circulant matrices C(Z) which entries are third order linear recurrent sequences. In addition, we compute eigenvalues, spectral norm and determinant of this matrix. Consequently, by using properties of this sequence, we obtain the eigenvalues, norms and determinants of circulant matrices with Cordonnier, Perrin and Van Der Laan numbers.

The cross-correlation function of complications of linear recurrent sequences

Abstract The paper is concerned with complications of linear recurrent sequences over the field GF(q) and the ring GR(qn , pn ) with interconnected recurrent relations. The cross-correlation function between cycles of given sequences is estimated.

Summations of linear recurrent sequences

We give an extension of Sister Celine’s method of proving hypergeometric sum identities that allows it to handle a larger variety of input summands. In particular, we extend the summand to powers of a C-finite sequence times a hypergeometric term. We then apply this to several problems. Some of these applications give new results, and some reprove already known results in an automated way.

Intervals without primes near elements of linear recurrence sequences

Let [Formula: see text] be an unbounded sequence of integers satisfying a linear recurrence relation with integer coefficients. We show that for any [Formula: see text] there exist infinitely many [Formula: see text] for which [Formula: see text] consecutive integers [Formula: see text] are all divisible by certain primes. Moreover, if the sequence of integers [Formula: see text] satisfying a linear recurrence relation is unbounded and non-degenerate then for some constant [Formula: see text] the intervals [Formula: see text] do not contain prime numbers for infinitely many [Formula: see text]. Applying this argument to sequences of integer parts of powers of Pisot and Salem numbers [Formula: see text] we derive a similar result for those sequences as well which implies, for instance, that the shifted integer parts [Formula: see text], where [Formula: see text] and [Formula: see text] runs through some infinite arithmetic progression of positive integers, are all composite.

The regularity of Diophantine quadruples

A set of positive integers is called a Diophantine tuple if the product of any two elements in the set increased by the unity is a perfect square. A conjecture on the regularity of Diophantine quadruples asserts that any Diophantine triple can be uniquely extended to a Diophantine quadruple by joining an element exceeding the maximal element of the triple. The problem is reduced to studying an equation expressed as the coincidence of two linear recurrence sequences with initial terms composed of the fundamental solutions of some Pellian equations. In this paper, we determine the values of those initial terms completely and obtain finiteness results on the number of solutions of the equation. As one of the applications to the problem on the regularity of Diophantine quadruples, we show in general that the number of ways of extending any given Diophantine triple is at most $11$.

Mod $p$ equivalence classes of linear recurrence sequences of degree~$2$

Laxton introduced a group structure on the set of equivalence classes of linear recurrence sequences of degree~2. This result yields much information on the divisibilities of such sequences. In this paper, we introduce other equivalence relations for the set of linear recurrence sequences $(G_n)$, which are defined by $G_0, G_1 \in \mathbb{Z} $ and $G_n=TG_{n-1}-NG_{n-2}$ for fixed integers~$T$ and $N=\pm 1$. The relations are given by certain congruences modulo~$p$ for a fixed prime number~$p$, which are different from Laxton's without modulo $p$ equivalence relations. We determine the initial terms $G_0$ and $G_1$ of all of the representatives of the equivalence classes $\overline {(G_n)}$ satisfying $p\nmid G_n$ for any integer~$n$ and give the number of equivalence classes. Furthermore, we determine the representatives of Laxton's without modulo~$p$ classes from our modulo~$p$ classes.

Weak and strong orders of linear recurring sequences

The concept of the strong order of linear recurring sequence (LRS) is introduced in this paper. Necessary and sufficient conditions for the existence of the strong LRS order are derived. The strong LRS order is exploited for the formalization of the problem of the extension of a sequence from the available fragment (fragments) of that sequence. The definition of the strong LRS order opens new possibilities for formal sequence analysis whenever the weak LRS order of that sequence exists. Computational experiments with discrete iterative maps are used to illustrate the applicability of the strong LRS order in nonlinear system analysis.

Encoding Trees by Linear Recurrence Sequences

A unified encoding of ordered binary trees with integer-valued labels at their vertices is proposed using linear forms of neighboring members of linear recursive sequences of the form Pn + 2 = an + 2 Pn + 1 + P, where P1 = P2 = 1; a3 and a4 … are natural numbers. Encoding and decoding procedures are simply implemented and use the recursive technique of direct pre-order depth-first tree traversal. A brief review of possible applications of the proposed encoding in problems of tree processing and cryptographic symmetric encryption is presented.

Image hiding in dynamic unstable self-organizing patterns

A digital image hiding scheme based on the breakup of spiral waves is presented in this paper. This scheme does not require initial conditions perturbation and embedding of the secret image is done during the evolution of a self-organizing pattern. Such features increase the security, but still enable an effective decoding of the secret image. The concept of the order of a 2D linear recurrent sequences are used to estimate the complexity of the pattern and select the optimal timing required for the pattern to complete. Computational experiments are used to demonstrate the properties and efficiency of the proposed scheme.

On a variant of Pillai's problem II

In this paper, we show that there are only finitely many c such that the equation Un−Vm=c has at least two distinct solutions (n,m), where {Un}n⩾0 and {Vm}m⩾0 are given linear recurrence sequences.

Estimating the number of solutions of systems of nonlinear equations with linear recurring arguments by the spectral method

Abstract A general approach is proposed for obtaining estimates of the number of solutions of systems of nonlinear equations. Final estimates are established in the case when the arguments of functions in the system are the signs of linear recurrent sequences over Galois rings.

On monochromatic linear recurrence sequences

In this paper we prove some van der Waerden type theorems for linear recurrence sequences. Under the assumption $a_{i-1}\leq a_{i}a_{s-1}$ ($i=2,\ldots,s$), we extend the results of G. Nyul and B. Rauf \cite{nyul} for sequences satisfying $x_i=a_1x_{i-s}+\ldots+a_sx_{i-1}$ ($i\geq s+1$), where $a_{1},\ldots,a_{s}$ are positive integers. Moreover, we solve completely the same problem for sequences satisfying the binary recurrence relation $x_i=ax_{i-1}-bx_{i-2}$ ($i\geq 3$) and $x_1<x_2$, where $a,b$ are positive integers with $a\geq b+1$.

Quadratic form Approach for the Number of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

We consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on an irreducible characteristic polynomial of degree $n$ and order $m$. Let $t=(q^{n}-1)/ m$. We use quadratic forms over finite fields to give the exact number of occurrences of zeros of the sequence within its least period when $t$ has q-adic weight 2. Consequently we prove that the cardinality of the set of zeros for sequences from this category is equal to two.

The Cardinality of the Set of Zeros of Homogeneous Linear Recurring Sequences over Finite Fields

Consider homogeneous linear recurring sequences over a finite field $\mathbb{F}_{q}$, based on the irreducible characteristic polynomial of degree $d$ and order $m$. We give upper and lower bounds, and in some cases the exact values of the cardinality of the set of zeros of the sequences within its least period. We also prove that the cyclotomy bound introduced here is the best upper bound as it is reached in infinitely many cases. In addition, the exact number of occurrences of zeros is determined using the correlation with irreducible cyclic codes when $(q^{d}-1)/ m$ follows the quadratic residue conditions and also when it has the form $q^{2a}-q^{a}+1$ where $a\in \mathbb{N}$.

Linear combinations of prime powers in binary recurrence sequences

We give finiteness results concerning terms of linear recurrence sequences having a representation as a linear combination, with fixed coefficients, of powers of fixed primes. On one hand, under certain conditions, we give effective bounds for the terms of binary recurrence sequences with such a representation. On the other hand, in the case of some special binary recurrence sequences, all terms having a representation as sums of powers of [Formula: see text] and [Formula: see text] are explicitly determined.