scholarly journals On Diophantine Equations Related to Order of Appearance in Fibonacci Sequence

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1073 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Prime Numbers ◽  
All Solutions ◽  
Adic Valuation

Let F n be the nth Fibonacci number. Order of appearance z ( n ) of a natural number n is defined as smallest natural number k, such that n divides F k . In 1930, Lehmer proved that all solutions of equation z ( n ) = n ± 1 are prime numbers. In this paper, we solve equation z ( n ) = n + ℓ for | ℓ | ∈ { 1 , … , 9 } . Our method is based on the p-adic valuation of Fibonacci numbers.

Representing Numbers Using Fibonacci Variants

2015 ◽  
Author(s):  
Stephen K. Lucas
Keyword(s):  
Natural Number ◽  
Greedy Algorithm ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Fixed Number ◽  
Natural Numbers ◽  
Form Versus ◽  
Entire Number

This chapter introduces the Zeckendorf representation of a Fibonacci sequence, a form of a natural number which can be easily found using a greedy algorithm: given a number, subtract the largest Fibonacci number less than or equal to it, and repeat until the entire number is used up. This chapter first compares the efficiency of representing numbers using Zeckendorf form versus traditional binary with a fixed number of digits and shows when Zeckendorf form is to be preferred. It also shows what happens when variants of Zeckendorf form are used. Not only can natural numbers as be presented sums of Fibonacci numbers, but arithmetic can also be done with them directly in Zeckendorf form. The chapter includes a survey of past approaches to Zeckendorf representation arithmetic, as well as some improvements.

scholarly journals Some Diophantine Problems Related to k-Fibonacci Numbers

Mathematics ◽  
2020 ◽  
Vol 8 (7) ◽  
pp. 1047
Author(s):  
Pavel Trojovský ◽  
Štěpán Hubálovský
Keyword(s):  
Recurrence Relation ◽  
Diophantine Equations ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Recursive Relation ◽  
Initial Values ◽  
Lucas Sequence ◽  
Diophantine Problems ◽  
Positive Integers

Let k ≥ 1 be an integer and denote ( F k , n ) n as the k-Fibonacci sequence whose terms satisfy the recurrence relation F k , n = k F k , n − 1 + F k , n − 2 , with initial conditions F k , 0 = 0 and F k , 1 = 1 . In the same way, the k-Lucas sequence ( L k , n ) n is defined by satisfying the same recursive relation with initial values L k , 0 = 2 and L k , 1 = k . The sequences ( F k , n ) n ≥ 0 and ( L k , n ) n ≥ 0 were introduced by Falcon and Plaza, who derived many of their properties. In particular, they proved that F k , n 2 + F k , n + 1 2 = F k , 2 n + 1 and F k , n + 1 2 − F k , n − 1 2 = k F k , 2 n , for all k ≥ 1 and n ≥ 0 . In this paper, we shall prove that if k > 1 and F k , n s + F k , n + 1 s ∈ ( F k , m ) m ≥ 1 for infinitely many positive integers n, then s = 2 . Similarly, that if F k , n + 1 s − F k , n − 1 s ∈ ( k F k , m ) m ≥ 1 holds for infinitely many positive integers n, then s = 1 or s = 2 . This generalizes a Marques and Togbé result related to the case k = 1 . Furthermore, we shall solve the Diophantine equations F k , n = L k , m , F k , n = F n , k and L k , n = L n , k .

scholarly journals On Integer Numbers with Locally Smallest Order of Appearance in the Fibonacci Sequence

2011 ◽  
Vol 2011 ◽  
pp. 1-4 ◽  
Author(s):  
Diego Marques
Keyword(s):  
Natural Number ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Natural Numbers

LetFnbe thenth Fibonacci number. The order of appearancez(n)of a natural numbernis defined as the smallest natural numberksuch thatndividesFk. For instance, for alln=Fm≥5, we havez(n−1)>z(n)<z(n+1). In this paper, we will construct infinitely many natural numbers satisfying the previous inequalities and which do not belong to the Fibonacci sequence.

PERFECT POWERS THAT ARE SUMS OF TWO POWERS OF FIBONACCI NUMBERS

2018 ◽  
Vol 99 (1) ◽  
pp. 34-41
Author(s):  
ZHONGFENG ZHANG ◽  
ALAIN TOGBÉ
Keyword(s):  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Image Position ◽  
Positive Integers ◽  
Complete Solutions

In this paper, we consider the Diophantine equations $$\begin{eqnarray}\displaystyle F_{n}^{q}\pm F_{m}^{q}=y^{p} & & \displaystyle \nonumber\end{eqnarray}$$ with positive integers $q,p\geq 2$ and $\gcd (F_{n},F_{m})=1$, where $F_{k}$ is a Fibonacci number. We obtain results for $q=2$ or $q$ an odd prime with $q\equiv 3\;(\text{mod}\;4),3<q<1087$, and complete solutions for $q=3$.

The Diophantine equation Fn = P(x)

2020 ◽  
Vol 16 (09) ◽  
pp. 2095-2111
Author(s):  
Szabolcs Tengely ◽  
Maciej Ulas
Keyword(s):  
Diophantine Equation ◽  
Diophantine Equations ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Infinitely Many Solutions ◽  
Polynomial Formula ◽  
Positive Integers ◽  
Integral Solutions

We consider equations of the form [Formula: see text], where [Formula: see text] is a polynomial with integral coefficients and [Formula: see text] is the [Formula: see text]th Fibonacci number that is, [Formula: see text] and [Formula: see text] for [Formula: see text] In particular, for each [Formula: see text], we prove the existence of a polynomial [Formula: see text] of degree [Formula: see text] such that the Diophantine equation [Formula: see text] has infinitely many solutions in positive integers [Formula: see text]. Moreover, we present results of our numerical search concerning the existence of even degree polynomials representing many Fibonacci numbers. We also determine all integral solutions [Formula: see text] of the Diophantine equations [Formula: see text] for [Formula: see text] and [Formula: see text].

The order of appearance of the sum and difference between two Fibonacci numbers

2019 ◽  
Vol 12 (03) ◽  
pp. 1950046 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Natural Number ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Greatest Common Divisor ◽  
Lucas Number ◽  
Number Formula

Let [Formula: see text] be the [Formula: see text]th Fibonacci number and [Formula: see text] be the [Formula: see text]th Lucas number. The order of appearance [Formula: see text] of a natural number [Formula: see text] is defined as the smallest natural number [Formula: see text] such that [Formula: see text] divides [Formula: see text]. For instance, [Formula: see text], for all [Formula: see text]. In this paper, among other things, we prove that [Formula: see text] depends on [Formula: see text], where [Formula: see text] is the greatest common divisor of numbers [Formula: see text] and [Formula: see text], which fulfill the condition [Formula: see text].

scholarly journals On the problem of Pillai with k-generalized Fibonacci numbers and powers of 3

2020 ◽  
Vol 16 (07) ◽  
pp. 1643-1666
Author(s):  
Mahadi Ddamulira ◽  
Florian Luca
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Fibonacci Sequence ◽  
Generalized Fibonacci Numbers ◽  
Generalized Fibonacci Sequence ◽  
Tribonacci Numbers

For an integer [Formula: see text], let [Formula: see text] be the [Formula: see text]-generalized Fibonacci sequence which starts with [Formula: see text] (a total of [Formula: see text] terms) and for which each term afterwards is the sum of the [Formula: see text] preceding terms. In this paper, we find all integers [Formula: see text] with at least two representations as a difference between a [Formula: see text]-generalized Fibonacci number and a power of [Formula: see text]. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbers.

THE FIBONACCI-NORM OF A POSITIVE INTEGER: OBSERVATIONS AND CONJECTURES

2010 ◽  
Vol 06 (02) ◽  
pp. 371-385 ◽  
Author(s):  
JEONG SOON HAN ◽  
HEE SIK KIM ◽  
J. NEGGERS
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Natural Numbers

In this paper, we define the Fibonacci-norm [Formula: see text] of a natural number n to be the smallest integer k such that n|Fk, the kth Fibonacci number. We show that [Formula: see text] for m ≥ 5. Thus by analogy we say that a natural number n ≥ 5 is a local-Fibonacci-number whenever [Formula: see text]. We offer several conjectures concerning the distribution of local-Fibonacci-numbers. We show that [Formula: see text], where [Formula: see text] provided Fm+k ≡ Fm (mod n) for all natural numbers m, with k ≥ 1 the smallest natural number for which this is true.

scholarly journals On Fibonacci Numbers of Order r Which Are Expressible as Sum of Consecutive Factorial Numbers

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 962
Author(s):  
Eva Trojovská  ◽  
Pavel Trojovský
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Initial Values

Let (tn(r))n≥0 be the sequence of the generalized Fibonacci number of order r, which is defined by the recurrence tn(r)=tn−1(r)+⋯+tn−r(r) for n≥r, with initial values t0(r)=0 and ti(r)=1, for all 1≤i≤r. In 2002, Grossman and Luca searched for terms of the sequence (tn(2))n, which are expressible as a sum of factorials. In this paper, we continue this program by proving that, for any ℓ≥1, there exists an effectively computable constant C=C(ℓ)>0 (only depending on ℓ), such that, if (m,n,r) is a solution of tm(r)=n!+(n+1)!+⋯+(n+ℓ)!, with r even, then max{m,n,r}<C. As an application, we solve the previous equation for all 1≤ℓ≤5.

scholarly journals On the Sum of Powers of Two k-Fibonacci Numbers which Belongs to the Sequence of k-Lucas Numbers

2016 ◽  
Vol 67 (1) ◽  
pp. 41-46
Author(s):  
Pavel Trojovský
Keyword(s):  
Recurrence Relation ◽  
Initial Conditions ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Lucas Numbers ◽  
Initial Values ◽  
Lucas Sequence ◽  
Positive Integers

Abstract Let k ≥ 1 and denote (Fk,n)n≥0, the k-Fibonacci sequence whose terms satisfy the recurrence relation Fk,n = kFk,n−1 +Fk,n−2, with initial conditions Fk,0 = 0 and Fk,1 = 1. In the same way, the k-Lucas sequence (Lk,n)n≥0 is defined by satisfying the same recurrence relation with initial values Lk,0 = 2 and Lk,1 = k. These sequences were introduced by Falcon and Plaza, who showed many of their properties, too. In particular, they proved that Fk,n+1 + Fk,n−1 = Lk,n, for all k ≥ 1 and n ≥ 0. In this paper, we shall prove that if k ≥ 1 and $F_{k,n + 1}^s + F_{k,n - 1}^s \in \left( {L_{k,m} } \right)_{m \ge 1} $ for infinitely many positive integers n, then s =1.

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