Linear Recurrence Sequences Research Articles

The number of appearances of elements on segments of linear recurrent sequences over Galois rings

Получены оценки для частот появлений элементов на отрезках линейных рекуррентных последовательностей над кольцами Галуа при условии, что начальный вектор рекуррентных последовательностей выбирается случайно и равновероятно.

On the Diophantine equation 𝑈_{𝑛}-𝑏^{𝑚}=𝑐

Let ( U n ) n ∈ N (U_n)_{n\in \mathbb {N}} be a fixed linear recurrence sequence defined over the integers (with some technical restrictions). We prove that there exist effectively computable constants B B and N 0 N_0 such that for any b , c ∈ Z b,c\in \mathbb {Z} with b > B b> B the equation U n − b m = c U_n - b^m = c has at most two distinct solutions ( n , m ) ∈ N 2 (n,m)\in \mathbb {N}^2 with n ≥ N 0 n\geq N_0 and m ≥ 1 m\geq 1 . Moreover, we apply our result to the special case of Tribonacci numbers given by T 1 = T 2 = 1 T_1= T_2=1 , T 3 = 2 T_3=2 and T n = T n − 1 + T n − 2 + T n − 3 T_{n}=T_{n-1}+T_{n-2}+T_{n-3} for n ≥ 4 n\geq 4 . By means of the LLL-algorithm and continued fraction reduction we are able to prove N 0 = 2 N_0=2 and B = e 438 B=e^{438} . The corresponding reduction algorithm is implemented in Sage.

On the size of a linear combination of two linear recurrence sequences over function fields

Let G_n and H_m be two non-degenerate linear recurrence sequences defined over a function field F in one variable over mathbb {C}, and let mu be a valuation on F. We prove that under suitable conditions there are effectively computable constants c_1 and C' such that the bound μ(Gn-Hm)≤μ(Gn)+C′\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\mu (G_n - H_m) \\le \\mu (G_n) + C' \\end{aligned}$$\\end{document}holds for max left( n,m right) > c_1 .

Additive Diophantine Equations with Binary Recurrences, {{mathcal {S}}}-Units and Several Factorials

There are many results in the literature concerning linear combinations of factorials among terms of linear recurrence sequences. Recently, Grossman and Luca provided effective bounds for such terms of binary recurrence sequences. In this paper we show that under certain conditions, even the greatest prime divisor of u_n-a_1m_1!-dots -a_km_k! tends to infinity, in an effective way. We give some applications of this result, as well.

On families of cubic split Thue equations parametrised by linear recurrence sequences

On families of cubic split Thue equations parametrised by linear recurrence sequences

Cullen Numbers in Sums of Terms of Recurrence Sequence

Let $$(U_n)_{n\ge 0}$$ be a fixed linear recurrence sequence of integers with order at least two, and for any positive integer $$\ell $$ , let $$\ell \cdot 2^{\ell } + 1$$ be a Cullen number. Recently in Bilu et al. (J Number Theory 202:412–425, 2019), generalized Cullen numbers in terms of linear recurrence sequence $$(U_n)_{n\ge 0}$$ under certain weak assumptions has been studied. In this paper, we consider the more general Diophantine equation $$U_{n_1} + \cdots + U_{n_k} = \ell \cdot x^{\ell } + Q(x)$$ , for a given polynomial $$Q(x) \in \mathbb {Z}[x]$$ and prove an effective finiteness result. Furthermore, we demonstrate our method by an example.

Representations of skew linear recurrent sequences of maximal period over finite field

Пусть $p$ - простое число, $R=\mathrm{GF}(q)$ - поле из $q$ элементов, где $q = p^r$, $S=\mathrm{GF}(q^{n})$ - его расширение степени $n$ и $\check{S}$ - кольцо линейных преобразований векторного пространства $_RS$. Последовательность $v$ над $S$, удовлетворяющую закону рекурсии $$ \forall i\in\mathbb{N}_0\colon v(i+m)= \psi_{m-1}(v(i+m-1))+\ldots+\psi_0(v(i)),\psi_0,\ldots,\psi_{m-1 }\in\check{S}, $$ будем называть скрученной линейной рекуррентной последовательностью над $S$ порядка $m$ с характеристическим многочленом $\Psi(x) = x^m - \sum_{j=0}^{m-1}\psi_jx^j$. Максимально возможный период последовательности такого вида равен $ q^{mn}-1$. Пусть $v$ - скрученная ЛРП максимального периода над $S$. Далее для произвольного кольца $J$ с единицей $\mathbf{e}$, для которого элемент $q\mathbf{e}$ не является делителем нуля, и отображения $f: S \to J$ при некоторых условиях описан аннулятор последовательности $f(v)$.

A general criterion for the Pólya-Carlson dichotomy and application

We prove a general criterion for an irrational power series f ( z ) = ∑ n = 0 ∞ a n z n f(z)=\sum _{n=0}^{\infty }a_nz^n with coefficients in a number field K K to admit the unit circle as a natural boundary. As an application, let F F be a finite field, let d d be a positive integer, let A ∈ M d ( F [ t ] ) A\in M_d(F[t]) be a d × d d\times d -matrix with entries in F [ t ] F[t] , and let ζ A ( z ) \zeta _A(z) be the Artin-Mazur zeta function associated to the multiplication-by- A A map on the compact abelian group F ( ( 1 / t ) ) d / F [ t ] d F((1/t))^d/F[t]^d . We provide a complete characterization of when ζ A ( z ) \zeta _A(z) is algebraic and prove that it admits the circle of convergence as a natural boundary in the transcendence case. This is in stark contrast to the case of linear endomorphisms on R d / Z d \mathbb {R}^d/\mathbb {Z}^d in which Baake, Lau, and Paskunas [Monatsh. Math. 161 (2010), pp. 33–42] prove that the zeta function is always rational. Some connections to earlier work of Bell, Byszewski, Cornelissen, Miles, Royals, and Ward are discussed. Our method uses a similar technique in recent work of Bell, Nguyen, and Zannier [Amer. Math. Soc. 373 (2020), pp. 4889–4906] together with certain patching arguments involving linear recurrence sequences.

On the size of roots of a family of polynomials related to linear recurrence sequences

On the size of roots of a family of polynomials related to linear recurrence sequences

-ADIC QUOTIENT SETS: LINEAR RECURRENCE SEQUENCES

AbstractLet $(x_n)_{n\geq 0}$ be a linear recurrence of order $k\geq 2$ satisfying $x_n=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$ for all integers $n\geq k$ , where $a_1,\ldots ,a_k,x_0,\ldots , x_{k-1}\in \mathbb {Z},$ with $a_k\neq 0$ . Sanna [‘The quotient set of k-generalised Fibonacci numbers is dense in $\mathbb {Q}_p$ ’, Bull. Aust. Math. Soc.96(1) (2017), 24–29] posed the question of classifying primes p for which the quotient set of $(x_n)_{n\geq 0}$ is dense in $\mathbb {Q}_p$ . We find a sufficient condition for denseness of the quotient set of the kth-order linear recurrence $(x_n)_{n\geq 0}$ satisfying $ x_{n}=a_1x_{n-1}+a_2x_{n-2}+\cdots +a_kx_{n-k}$ for all integers $n\geq k$ with initial values $x_0=\cdots =x_{k-2}=0,x_{k-1}=1$ , where $a_1,\ldots ,a_k\in \mathbb {Z}$ and $a_k=1$ . We show that, given a prime p, there are infinitely many recurrence sequences of order $k\geq 2$ whose quotient sets are not dense in $\mathbb {Q}_p$ . We also study the quotient sets of linear recurrence sequences with coefficients in certain arithmetic and geometric progressions.

Using irreducible polynomials for random number generation

A method is presented for generating random numbers with uniform distribution using linear recurrence sequences with very large period lengths. This method requires an irreducible polynomial modulo 2 to define the sequence. A suitable method for generating an infinite number of such polynomials is presented. The polynomials generated in this way can have an arbitrarily large degree, and a large enough order to make them suitable for practical applications.

On the reciprocal sums of products of $ m $th-order linear recurrence sequences

<abstract><p>In this paper, we use the method of error estimation to consider the reciprocal sums of products of any $ m $th-order linear recurrence sequences $ \left \{ u_{n} \right \} $. Specifically, we find that a series of sequences are "asymptotically equivalent" to the reciprocal sums of products of any $ m $th-order linear recurrence sequences $ \left \{ u_{n} \right \} $.</p></abstract>

Connecting slow solutions to nested recurrences with linear recurrent sequences

Labeled infinite trees provide combinatorial interpretations for many integer sequences generated by nested recurrence relations. Typically, such sequences are monotone increasing. Several of these sequences also have straightforward descriptions in terms of how often each value in the sequence occurs. In this paper, we generalize the most classical examples to a larger family of sequences parametrized by linear recurrence relations. Each of our sequences can be constructed in three different ways: via a nested recurrence relation, from labeled infinite trees, or by using Zeckendorf-like strings of digits to describe its frequency sequence. We conclude the paper by discussing the asymptotic behaviors of our sequences.

Consecutive Ratios in Second-Order Linear Recurrence Sequences

Abstract Let (a n )n=0 ∞ be a second-order linear recurrence sequence with constant coefficient. We study the limit points and asymptotic distribution of the sequence of consecutive ratios a n +1/a n .

S-parts of sums of terms of linear recurrence sequences

Let \(S= \{ p_1, \ldots, p_s\}\) be a finite, non-empty set of distinct prime numbers and \((U_{n})_{n \geq 0}\) be a linear recurrence sequence of integers of order at least 2. For any positive integer k, and \(w = (w_k, \ldots, w_1)\in\mathbb{Z}^k, w_1, \ldots, w_k\neq 0\) we define \((U_j^{(k, w)})_{j\geq 1}\) an increasing sequence composed of integers of the form \(|w_kU_{n_k} +\cdots + w_1U_{n_1}|\), \( n_k>\cdots >n_1\). Under certain assumptions, we prove that for any \(\varepsilon >0\), there exists an integer \(n_{0}\) such that \([U_j^{(k,w)}]_S < (U_j^{(k, w)})^{\varepsilon},\) \({\rm for}\, j > n_0\), where \([m]_S\) denotes the S-part of the positive integer m. On further assumptions on \((U_{n})_{n \geq 0}\), we also compute an effective bound for \([U_j^{(k, w)}]_S\) of the form \((U_j^{(k,w)})^{1-c}\), where c is a positive constant depending only on r, \(a_1\) , . . ., \(a_r\), \(U_0\), . . ., \(U_{r-1}\) , \(w_1\), . . ., \(w_k\) and S.

The level of distribution of the sum-of-digits function of linear recurrence number systems

Let $G=(G_j)_{j\ge 0}$ be a strictly increasing linear recurrent sequence of integers with $G_0=1$ having characteristic polynomial $X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$. It is well known that each positive integer $\nu$ can be uniquely represented by the so-called greedy expansion $\nu=\varepsilon_0(\nu)G_0+\cdots+\varepsilon_\ell(\nu)G_\ell$ for $\ell \in \mathbb{N}$ satisfying $G_\ell \le \nu 0}$ that \[ \sum_{q<x^{\vartheta-\varepsilon}}\max_{z<x}\max_{1\leq h\leq q} \lvert\sum_{\substack{k<z,s_G(k)\equiv r\bmod s k\equiv h\bmod q}}1 -\frac1q\sum_{k<z,s_G(k)\equiv r\bmod s}1\rvert \ll x(\log 2x)^{-A}. \] Here $\vartheta=\vartheta(G) \ge \frac12$ can be computed explicitly and we have $\vartheta(G) \to 1$ for $a_1\to\infty$. As an application we show that $\#\{ k\le x \,:\, s_G(k) \equiv r \pmod{s}, \; k \hbox{ has at most two prime factors} \} \gg x/\log x $ provided that the coefficient $a_1$ is not too small. Moreover, using Bombieri's sieve an almost prime number theorem for $s_G$ follows from our result. Our work extends earlier results on the classical $q$-ary sum-of-digits function obtained by Fouvry and Mauduit.

A robust class of linear recurrence sequences

We introduce a subclass of linear recurrence sequences which we call poly-rational sequences because they are denoted by rational expressions closed under sum and product. We show that this class is robust by giving several characterisations: polynomially ambiguous weighted automata, copyless cost-register automata, rational formal series, and linear recurrence sequences whose eigenvalues are roots of rational numbers.

Bent functions in the partial spread class generated by linear recurring sequences

We present a construction of partial spread bent functions using subspaces generated by linear recurring sequences (LRS). We first show that the kernels of the linear mappings defined by two LRS have a trivial intersection if and only if their feedback polynomials are relatively prime. Then, we characterize the appropriate parameters for a family of pairwise coprime polynomials to generate a partial spread required for the support of a bent function, showing that such families exist if and only if the degrees of the underlying polynomials are either 1 or 2. We then count the resulting sets of polynomials and prove that, for degree 1, our LRS construction coincides with the Desarguesian partial spread. Finally, we perform a computer search of all mathcal{PS}mathcal{}^- and mathcal{PS}mathcal{}^+ bent functions of n=8 variables generated by our construction and compute their 2-ranks. The results show that many of these functions defined by polynomials of degree d=2 are not EA-equivalent to any Maiorana–McFarland or Desarguesian partial spread function.

Ruin probability for finite negative binomial mixture claims via recurrence sequences

A new procedure to find the ultimate ruin probability in a discrete-time risk model is presented for claims with a mixture of m negative binomial distributions. The method involves the theory of linear recurrence sequences. It requires to find the zeroes of an m degree polynomial and the solution of a system of m linear equations. Numerical results and plots are provided as examples.

Linear recurrent cryptography: Golden-like cryptography for higher order linear recurrences

We develop matrix cryptography based on linear recurrent sequences of any order that allows securing encryption against brute force and chosen plaintext attacks. In particular, we solve the problem of generalizing error detection and correction algorithms of golden cryptography previously known only for recurrences of a special form. They are based on proving the checking relations (inequalities satisfied by the ciphertext) under the condition that the analog of the golden Q-matrix has the strong Perron–Frobenius property. These algorithms are proved to be especially efficient when the characteristic polynomial of the recurrence is a Pisot polynomial. Finally, we outline algorithms for generating recurrences that satisfy our conditions.