Linear combinations of prime powers in binary recurrence sequences

A balancing-like sequence is a binary recurrence sequence which generalizes the balancing sequence and the sequence of nonnegative integers. This sequence, under certain assumptions, may be used to describe the growth of fortune of a person engaged in some business or profession. Since, in any business or profession, the growth is influenced by many uncertainties, it is more natural to induce some sort of randomness in the balancing-like sequences. If, in a balancing-like sequence, the growth rate is assumed to be a random variable, the resulting sequence will be a stochastic process and the sequence of expectations, in many cases, cannot be described as a binary recurrence sequence. In some cases, the growth rate of expectations increases without limit while, in some cases, it remains finite.

In this paper, we look at the problem of expressing a term of a given nondegenerate binary recurrence sequence as a linear combination of a factorial and an \(S\)-unit whose coefficients are bounded. In particular, we find the largest member of the Fibonacci sequence which can be written as a sum or a difference between a factorial and an \(S\)-unit associated to the set of primes \(\{2,3,5,7\}\).

On binary recurrence sequences

A Characterization of Certain Binary Recurrence Sequences

Let $(x_n)_{n\geq 0}$ be a linear recurrence sequence of order $k\geq 2$ satisfying $$ \begin{align*}x_n=a_1x_{n-1}+a_2x_{n-2}+\dots+a_kx_{n-k}\end{align*} $$ for all integers $n\geq k$ , where $a_1,\dots ,a_k,x_0,\dots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . In 2017, Sanna posed an open question to classify primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In a recent paper, we showed that if the characteristic polynomial of the recurrence sequence has a root $\pm \alpha $ , where $\alpha $ is a Pisot number and if p is a prime such that the characteristic polynomial of the recurrence sequence is irreducible in $\mathbb {Q}_p$ , then the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . In this article, we answer the problem for certain linear recurrence sequences whose characteristic polynomials are reducible over $\mathbb {Q}$ .

Tropical recurrent sequences

Information systems are developing and improving very fast. The requirements for synchronization for such systems also increase. In such systems linear recurrent sequences are widely used for synchronization. In this paper filtration algorithms of linear recurrent sequences sequences are synthesized. The synthesized algorithms can be classified as quasi-noncoherent (subincoherent) algorithms, as opposed to quasi-coherent algorithms, which use phase estimate of the received signal, or to noncoherent algorithms that use averaging over the entire region of phase uncertainty. The results of computer simulation of developed algorithms for various signal-to-noise ratios and accumulation time are presented.

Linear orthogonal binary sequences, special M-Sequences, are used widely in the systems communication channels as in the forward links for mixing the information on connection and as in the backward links of these channels to sift this information which transmitted and the receivers get the information in a correct form In current research trying study the construction of the linear equivalent of a product sequence on two, three, and four degrees over a linear sequences from the field Fp, where p is larger than two, and answering on the request, how is the maximum length of the linear equivalent of a product sequence (on a linear sequence {an} over the field Fp), is it less than rNh as the binary sequences?, or can reach it ?, or the length is exceed this value rNh? And is the product sequences are orthogonal? And we show that in some cases, the maximum length rNh for the binary sequences is not correct for the linear sequences in the finite field Fp for p larger than two and the result product sequences are not orthogonal, also trying study the product sequence on two different LFSRs, and how can use one shift feedback shift register LFSR as a monitor register of other p registers. In the current time, I think, there is no coders or decoders using the sequences over finite fields Fp where p is larger than 2 and from this idea this article showing very need for using in the future.

A technique to study the correlation measures of binary sequences

Pure powers in recurrence sequences and some related diophantine equations

Compact or efficient representation for either images or image sequences is key operation to performing image and video processing tasks, such as compression, analysis, etc. The efficiency of an approximation is evaluated by the sparsity measure of the approximation, i.e., the sparsest the representation is, the more efficient it is. For image processing tasks, it is often desired to decompose the image into a linear combination of few visual primitives or features selected from a large collection of waveforms, called the dictionary. These primitives are usually designed in such a way to achieve some good approximation performances by assuming a given class of functions to model images. A common class of functions for image modeling is the set of functions having discontinuities along smooth contours or boundaries, delineating smooth geometrical regions. It was shown that the well-known separable isotropically scaled two-dimensional wavelets fail to capture the geometrical regularity inherent to images. Motivated by this issue, we designed two rich dictionaries Ds and Dst, composed of multi-scaled ridge-like functions satisfying the anisotropy and the directionality features, for image and video expansions, respectively. Regarding the spatio-temporal dictionary Dst, we defined a warping operator W, which aligns these functions along the coherent motion trajectories in order to exploit the nature of the temporal evolution in the video signal. To obtain a compact representation for the visual signal over the dictionaries Ds and Dst, only the primitives that best match the signal components must be selected. This problem is well studied in approximation theory and it is known as the problem of sparse approximation in unrestricted dictionaries, whose optimal solution is NP-hard. Greedy approaches, such as MP and OMP, have been proposed to provide sub-optimal solutions by iteratively choosing one atom at a time. However, the computational complexity of these algorithms is cumbersome and limits their applicability. To alleviate this issue, we introduced a greedy algorithm, called the M-Term Pursuit MTP, whose performances are very close to those related to MP. Then we employed both algorithms, MP and MTP, for image and image sequence decompositions and we investigated their performances. A target application has been studied, which is the scalable compression, where the aim is to generate a bit-stream that can provide both SNR and geometrical scalability. Geometrical scalability refers to the spatial scalability in case the of images and to the spatio-temporal scalability in the case of video sequences. The SNR or quality scalability is related mainly to the nature of the greedy approaches and the embedded quantization and coding. To do so, we designed a rate allocation algorithm that offers a progressively refinable bit-stream, based on the subsets approach. On the other hand, the geometrical scalability is fullfiled thanks to the structure of the dictionaries, Ds and Dst, which are designed using parametric fashion. Comparisons with state-of-the-art scalable and non-scalable codecs illustrate the good performance of the proposed compression techniques at low rates.

We prove that apart from explicitly given cases, described in terms of Dickson polynomials, a polynomial fin mathbb {Q}[x] can have at most one shift f(x)-lambda (lambda in mathbb {C}) of the form u(g(x))^q(h(x))^k with uin mathbb {C}, g,hin mathbb {C}[x] and either deg (g)=2, k is even, q=k/2 or deg (g)le 1, kge 2, qge 1. This is shown by handling the case of two possible shifts, which was an open issue. As an application, we give a precise statement yielding a description of polynomials f having infinitely many shifted power (S-integral) values, and a complete description of superelliptic equations having infinitely many S-integral solutions when the polynomial involved is composite. In the case where there are finitely many solutions, our results yield effective bounds for them. Finally, as further applications, we give effective results for polynomial values in the solutions of Pell equations and in non-degenerate binary recurrence sequences.

This paper examines tropical recurrent sequences associated with Somos sequences. The classical Somos sequences have applications in the theory of elliptic curves. Due to the Laurent character of the classical sequences, a pattern can be inferred between the classical sequences and their tropical counterparts. The greatest interest is the increase in the dimension of the solution space of tropical sequences depending on the length of the final sequence. For a set of tropical sequences described by tropical recurrence relations, D.Yu.Grigoriev put forward a hypothesis about the stabilization of the maximum dimensions of the components of the corresponding tropical prevarieties. This hypothesis has been proven for tropical linear recurrent sequences. As part of this work, for tropical recurrent sequences associated with the sequences Somos-4 and Somos-5, the corresponding tropical prevarieties were investigated using the Gfan package in order to test Grigoriev's hypothesis.

In this paper, closed forms of the summation formulas for generalized Tribonacci numbers are presented. Then, some previous results are recovered as particular cases of the present results. As special cases, we give summation formulas of Tribonacci, Tribonacci-Lucas, Padovan, Perrin, Narayana and some other third order linear recurrance sequences. All the summing formulas of well known recurrence sequences which we deal with are linear except the cases Pell-Padovan and Padovan-Perrin.

Multi-level signals and sequences have become a significant aspect of modern high-speed communication systems. Hence, to ensure the confidentiality and integrity of the transmitted information, advanced methods and devices are necessary to produce strong cryptographic properties for not only binary but also for nonbinary keystreams, which can be used in resource-constrained microcontrollers. The proposed method and apparatus generate both a balanced nonlinear 5-ary pseudo-random sequence and a binary keystream sequence. The nonlinearity is determined by applying shrinking and multiplexing techniques to the generated linear 5-ary pseudo-random sequence and the running zero in its conversion to a binary balanced sequence.