On the local behavior of the order of appearance in the Fibonacci sequence

2014 ◽  
Vol 10 (04) ◽  
pp. 915-933 ◽  
Author(s):  
Florian Luca ◽  
Carl Pomerance
Keyword(s):  
Positive Integer ◽  
Rational Function ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Asymptotic Density ◽  
Infinitely Many Solutions ◽  
Local Behavior ◽  
Sieve Methods ◽  
Density Zero

Let z(N) be the order of appearance of N in the Fibonacci sequence. This is the smallest positive integer k such that N divides the k th Fibonacci number. We show that each of the six total possible orderings among z(N), z(N + 1), z(N + 2) appears infinitely often. We also show that for each nonzero even integer c and many odd integers c the equation z(N) = z(N + c) has infinitely many solutions N, but the set of solutions has asymptotic density zero. The proofs use a result of Corvaja and Zannier on the height of a rational function at 𝒮-unit points as well as sieve methods.

scholarly journals On f(n) modulo Ω(n) and ω(n) when f is a polynomial

2004 ◽  
Vol 77 (2) ◽  
pp. 149-164 ◽  
Author(s):  
Florian Luca
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Density Zero ◽  
Prime Factors ◽  
Nonzero Polynomial

AbstractIn this paper we show that if f (X) ∈; Z [X ] is a nonzero polynomial, then ω(n)/f(n) holds only on a set of n of asymptotic density zero, where for a positive integer n the number ω(n) counts the number of distinct prime factors ofn.

scholarly journals On Two Problems Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3273
Author(s):  
Pavel Trojovský
Keyword(s):  
Functional Equation ◽  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Function ◽  
Fibonacci Sequence ◽  
Infinitely Many Solutions ◽  
Fermat's Last Theorem ◽  
Fermat’S Last Theorem ◽  
Divisibility Properties

The order of appearance (in the Fibonacci sequence) function z:Z≥1→Z≥1 is an arithmetic function defined for a positive integer n as z(n)=min{k≥1:Fk≡0(modn)}. A topic of great interest is to study the Diophantine properties of this function. In 1992, Sun and Sun showed that Fermat’s Last Theorem is related to the solubility of the functional equation z(n)=z(n2), where n is a prime number. In addition, in 2014, Luca and Pomerance proved that z(n)=z(n+1) has infinitely many solutions. In this paper, we provide some results related to these facts. In particular, we prove that limsupn→∞(z(n+1)−z(n))/(logn)2−ϵ=∞, for all ϵ∈(0,2).

scholarly journals The Proof of a Conjecture Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (20) ◽  
pp. 2638
Author(s):  
Eva Trojovská ◽  
Kandasamy Venkatachalam
Keyword(s):  
Positive Integer ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Divisibility Properties ◽  
Positive Integers

The order of appearance of n (in the Fibonacci sequence) z(n) is defined as the smallest positive integer k for which n divides the k—the Fibonacci number Fk. Very recently, Trojovský proved that z(n) is an even number for almost all positive integers n (in the natural density sense). Moreover, he conjectured that the same is valid for the set of integers n ≥ 1 for which the integer 4 divides z(n). In this paper, among other things, we prove that for any k ≥ 1, the number z(n) is divisible by 2k for almost all positive integers n (in particular, we confirm Trojovský’s conjecture).

scholarly journals Value distribution of meromorphic functions concerning rational functions and differences

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Mingliang Fang ◽  
Degui Yang ◽  
Dan Liu
Keyword(s):  
Positive Integer ◽  
Meromorphic Function ◽  
Rational Function ◽  
Finite Order ◽  
Meromorphic Functions ◽  
Rational Functions ◽  
Value Distribution ◽  
Exponent Of Convergence ◽  
Nonzero Constant ◽  
Zero Points

AbstractLet c be a nonzero constant and n a positive integer, let f be a transcendental meromorphic function of finite order, and let R be a nonconstant rational function. Under some conditions, we study the relationships between the exponent of convergence of zero points of $f-R$ f − R , its shift $f(z+nc)$ f ( z + n c ) and the differences $\Delta _{c}^{n} f$ Δ c n f .

scholarly journals A note on primes in certain residue classes

2018 ◽  
Vol 14 (08) ◽  
pp. 2219-2223
Author(s):  
Paolo Leonetti ◽  
Carlo Sanna
Keyword(s):  
Positive Integer ◽  
Asymptotic Density ◽  
Residue Classes ◽  
Euler Totient Function ◽  
The One ◽  
Positive Integers

Given positive integers [Formula: see text], we prove that the set of primes [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density relative to the set of all primes which is at least [Formula: see text], where [Formula: see text] is the Euler totient function. This result is similar to the one of Heilbronn and Rohrbach, which says that the set of positive integer [Formula: see text] such that [Formula: see text] for [Formula: see text] admits asymptotic density which is at least [Formula: see text].

scholarly journals ON THE DENSITY OF INTEGERS OF THE FORM (p−1)2−n IN ARITHMETIC PROGRESSIONS

2008 ◽  
Vol 78 (3) ◽  
pp. 431-436 ◽  
Author(s):  
XUE-GONG SUN ◽  
JIN-HUI FANG
Keyword(s):  
Positive Integer ◽  
Prime Number ◽  
Arithmetic Progression ◽  
Arithmetic Progressions ◽  
Asymptotic Density ◽  
Natural Numbers ◽  
Positive Proportion ◽  
Covering System

AbstractErdős and Odlyzko proved that odd integers k such that k2n+1 is prime for some positive integer n have a positive lower density. In this paper, we characterize all arithmetic progressions in which natural numbers that can be expressed in the form (p−1)2−n (where p is a prime number) have a positive proportion. We also prove that an arithmetic progression consisting of odd numbers can be obtained from a covering system if and only if those integers in such a progression which can be expressed in the form (p−1)2−n have an asymptotic density of zero.

ON UNIFORMLY RECURRENT MORPHIC SEQUENCES

2009 ◽  
Vol 20 (05) ◽  
pp. 919-940 ◽  
Author(s):  
FRANCOIS NICOLAS ◽  
YURI PRITYKIN
Keyword(s):  
Positive Integer ◽  
Computer Science ◽  
Polynomial Time ◽  
Linear Growth ◽  
Fibonacci Sequence ◽  
Theoretical Computer Science ◽  
Symbolic Sequence ◽  
Theoretical Computer ◽  
Automatic Sequences ◽  
The Status

A pure morphic sequence is a right-infinite, symbolic sequence obtained by iterating a letter-to-word substitution. For instance, the Fibonacci sequence and the Thue–Morse sequence, which play an important role in theoretical computer science, are pure morphic. Define a coding as a letter-to-letter substitution. The image of a pure morphic sequence under a coding is called a morphic sequence.A sequence x is called uniformly recurrent if for each finite subword u of x there exists an integer l such that u occurs in every l-length subword of x.The paper mainly focuses on the problem of deciding whether a given morphic sequence is uniformly recurrent. Although the status of the problem remains open, we show some evidence for its decidability: in particular, we prove that it can be solved in polynomial time on pure morphic sequences and on automatic sequences.In addition, we prove that the complexity of every uniformly recurrent, morphic sequence has at most linear growth: here, complexity is understood as the function that maps each positive integer n to the number of distinct n-length subwords occurring in the sequence.

Spiral growth inNephrolepidina: evidence of “golden selection”

Paleobiology ◽  
10.1666/12057 ◽  
2014 ◽  
Vol 40 (2) ◽  
pp. 151-161 ◽  
Author(s):  
Andrea Benedetti
Keyword(s):  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Spiral Growth ◽  
Regular Growth ◽  
Mean Values ◽  
Evolutionary Selection ◽  
Golden Angle ◽  
New Type ◽  
New Evidence ◽  
Interpretative Model

Examination of the neanic apparatuses of known populations ofNephrolepidina praemarginata,N. morgani, andN. tournouerireveals that the equatorial chamberlets are arranged in spirals, along the direction of connection of the oblique stolons, giving the optical effect of intersecting curves. InN. praemarginatacommonly 34 left- and right-oriented primary spirals occur from the first annulus to the periphery, 21 secondary spirals from the third to fifth annulus, 13 ternary spirals from the fifth to eighth annulus, following the Fibonacci sequence.The number of the spirals increases in larger specimens and in more embracing morphotypes, and especially in trybliolepidine specimens; the secondary and ternary spirals from the investigatedN. praemarginatatoN. tournoueripopulations tend to start from more distal annuli. An interpretative model of the spiral growth ofNephrolepidinais attempted.The angle formed by the basal annular stolon and distal oblique stolon in equatorial chamberlets ranges from 122° inN. praemarginatato mean values close to the golden angle (137.5°) inN. tournoueri.The increase in the Fibonacci number of spirals during the evolution of the lineage, along with the disposition of the stolons between contiguous equatorial chamberlets, provides new evidence of evolutionary selection for specimens with optimally packed chamberlets.Natural selection favors individuals with the most regular growth, which fills the equatorial space more efficiently, thus allowing these individuals to reach the adult stage faster. We refer to this new type of selection as “golden selection.”

scholarly journals Rational Divide-and-Conquer Relations

2011 ◽  
Vol 2011 ◽  
pp. 1-14
Author(s):  
Charinthip Hengkrawit ◽  
Vichian Laohakosol ◽  
Watcharapon Pimsert
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Rational Function ◽  
Natural Generalization ◽  
Divide And Conquer ◽  
Recursive Equation ◽  
Closed Form Solutions ◽  
Tangent Function

A rational divide-and-conquer relation, which is a natural generalization of the classical divide-and-conquer relation, is a recursive equation of the form f(bn)=R(f(n),f(n),…,f(b−1)n)+g(n), where b is a positive integer ≥2; R a rational function in b−1 variables and g a given function. Closed-form solutions of certain rational divide-and-conquer relations which can be used to characterize the trigonometric cotangent-tangent and the hyperbolic cotangent-tangent function solutions are derived and their global behaviors are investigated.

On the $x$-coordinates of Pell equations which are Fibonacci numbers

2018 ◽  
Vol 122 (1) ◽  
pp. 18 ◽  
Author(s):  
Florian Luca ◽  
Alain Togbé
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Pell Equation ◽  
Pell Equations

For an integer $d>2$ which is not a square, we show that there is at most one value of the positive integer $x$ participating in the Pell equation $x^2-dy^2=\pm 1$ which is a Fibonacci number.

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