Intervals without primes near elements of linear recurrence sequences

2018 ◽  
Vol 14 (02) ◽  
pp. 567-579
Author(s):  
Artūras Dubickas
Keyword(s):  
Recurrence Relation ◽  
Arithmetic Progression ◽  
Prime Numbers ◽  
Linear Recurrence ◽  
Integer Coefficients ◽  
Salem Numbers ◽  
Consecutive Integers ◽  
Linear Recurrence Relation ◽  
Recurrence Sequences ◽  
Positive Integers

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.

scholarly journals Matrices determined by a linear recurrence relation among entries

2003 ◽  
Vol 373 ◽  
pp. 89-99 ◽  
Author(s):  
Gi-Sang Cheon ◽  
Suk-Geun Hwang ◽  
Seog-Hoon Rim ◽  
Seok-Zun Song
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

scholarly journals Linear recurrences in the degree sequences of monomial mappings

2008 ◽  
Vol 28 (5) ◽  
pp. 1369-1375 ◽  
Author(s):  
ERIC BEDFORD ◽  
KYOUNGHEE KIM
Keyword(s):  
Recurrence Relation ◽  
Linear Recurrence ◽  
Degree Sequences ◽  
Integer Matrix ◽  
Linear Recurrences ◽  
Monomial Map ◽  
Linear Recurrence Relation

AbstractLet A be an integer matrix, and let fA be the associated monomial map. We give a connection between the eigenvalues of A and the existence of a linear recurrence relation in the sequence of degrees.

scholarly journals On linear recurrence sequences with polynomial coefficients

1996 ◽  
Vol 38 (2) ◽  
pp. 147-155 ◽  
Author(s):  
A. J. van der Poorten ◽  
I. E. Shparlinski
Keyword(s):  
Recurrence Relation ◽  
Upper Bound ◽  
Rational Numbers ◽  
Negative Integer ◽  
Linear Recurrence ◽  
Initial Values ◽  
Polynomial Coefficients ◽  
Image Position ◽  
Recurrence Sequences

We consider sequences (Ah)defined over the field ℚ of rational numbers and satisfying a linear homogeneous recurrence relationwith polynomial coefficients sj;. We shall assume without loss of generality, as we may, that the sj, are defined over ℤ and the initial values A0A]…, An−1 are integer numbers. Also, without loss of generality we may assume that S0 and Sn have no non-negative integer zero. Indeed, any other case can be reduced to this one by making a shift h → h – l – 1 where l is an upper bound for zeros of the corresponding polynomials (and which can be effectively estimated in terms of their heights)

General anharmonic oscillators

1978 ◽  
Vol 364 (1717) ◽  
pp. 265-275 ◽  
Keyword(s):  
Eigenvalue Problem ◽  
Asymptotic Behaviour ◽  
Recurrence Relation ◽  
Quantum Number ◽  
Anharmonic Oscillator ◽  
Linear Recurrence ◽  
Anharmonic Oscillators ◽  
Anharmonicity Constant ◽  
Linear Recurrence Relation

The eigenvalue problem of the general anharmonic oscillator (Hamiltonian H 2 μ ( k, λ ) = -d 2 / d x 2 + kx 2 + λx 2 μ , ( k, λ ) is investi­gated in this work. Very accurate eigenvalues are obtained in all régimes of the quantum number n and the anharmonicity constant λ . The eigenvalues, as functions of λ , exhibit crossings. The qualitative features of the actual crossing pattern are substantially reproduced in the W. K. B. approximation. Successive moments of any transition between two general anharmonic oscillator eigenstates satisfy exactly a linear recurrence relation. The asymptotic behaviour of this recursion and its consequences are examined.

Convolution Identities for Stirling Numbers of the First Kind

Integers ◽  
2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Takashi Agoh ◽  
Karl Dilcher
Keyword(s):  
Recurrence Relation ◽  
Stirling Numbers ◽  
Linear Recurrence ◽  
Linear Recurrence Relation

AbstractWe derive several new convolution identities for the Stirling numbers of the first kind. As a consequence we obtain a new linear recurrence relation which generalizes known relations.

scholarly journals A Reciprocity Theorem for Domino Tilings

10.37236/1562 ◽  
2001 ◽  
Vol 8 (1) ◽  
Author(s):  
James Propp
Keyword(s):  
Recurrence Relation ◽  
Reciprocity Theorem ◽  
Algebraic Methods ◽  
Linear Recurrence ◽  
Combinatorial Interpretation ◽  
Domino Tilings ◽  
Linear Recurrence Relation

Let $T(m,n)$ denote the number of ways to tile an $m$-by-$n$ rectangle with dominos. For any fixed $m$, the numbers $T(m,n)$ satisfy a linear recurrence relation, and so may be extrapolated to negative values of $n$; these extrapolated values satisfy the relation $$T(m,-2-n)=\epsilon_{m,n}T(m,n),$$ where $\epsilon_{m,n}=-1$ if $m \equiv 2$ (mod 4) and $n$ is odd and where $\epsilon_{m,n}=+1$ otherwise. This is equivalent to a fact demonstrated by Stanley using algebraic methods. Here I give a proof that provides, among other things, a uniform combinatorial interpretation of $T(m,n)$ that applies regardless of the sign of $n$.

scholarly journals A new method for solving the linear recurrence relation with nonhomogeneous constant coefficients in some cases

2021 ◽  
Vol 2113 (1) ◽  
pp. 012070
Author(s):  
Ben-Chao Yang ◽  
Xue-Feng Han
Keyword(s):  
Recurrence Relation ◽  
Computer Programming ◽  
Recursive Relation ◽  
New Method ◽  
Linear Recurrence ◽  
General Proof ◽  
Constant Coefficients ◽  
Linear Recurrence Relation ◽  
Almost All ◽  
General Method

Abstract Recursive relation mainly describes the unique law satisfied by a sequence, so it plays an important role in almost all branches of mathematics. It is also one of the main algorithms commonly used in computer programming. This paper first introduces the concept of recursive relation and two common basic forms, then starts with the solution of linear recursive relation with non-homogeneous constant coefficients, gives a new solution idea, and gives a general proof. Finally, through an example, the general method and the new method given in this paper are compared and verified.

scholarly journals On Sequences of Numbers and Polynomials Defined by Linear Recurrence Relations of Order 2

2009 ◽  
Vol 2009 ◽  
pp. 1-21 ◽  
Author(s):  
Tian-Xiao He ◽  
Peter J.-S. Shiue
Keyword(s):  
Recurrence Relation ◽  
Chebyshev Polynomials ◽  
Recurrence Relations ◽  
Linear Recurrence ◽  
Lucas Numbers ◽  
Polynomial Sequences ◽  
Fibonacci And Lucas Numbers ◽  
Linear Recurrence Relation ◽  
Humbert Polynomials ◽  
General Method

Here we present a new method to construct the explicit formula of a sequence of numbers and polynomials generated by a linear recurrence relation of order 2. The applications of the method to the Fibonacci and Lucas numbers, Chebyshev polynomials, the generalized Gegenbauer-Humbert polynomials are also discussed. The derived idea provides a general method to construct identities of number or polynomial sequences defined by linear recurrence relations. The applications using the method to solve some algebraic and ordinary differential equations are presented.

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