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.

scholarly journals Solution of certain Pell equations

2018 ◽  
Vol 11 (04) ◽  
pp. 1850056 ◽  
Author(s):  
Zahid Raza ◽  
Hafsa Masood Malik
Keyword(s):  
Positive Integer ◽  
Fundamental Solution ◽  
Positive Solutions ◽  
Continued Fraction ◽  
Pell Equation ◽  
Lucas Sequences ◽  
Pell Equations ◽  
Integer Solutions ◽  
Positive Integers

Let [Formula: see text] be any positive integers such that [Formula: see text] and [Formula: see text] is a square free positive integer of the form [Formula: see text] where [Formula: see text] and [Formula: see text] The main focus of this paper is to find the fundamental solution of the equation [Formula: see text] with the help of the continued fraction of [Formula: see text] We also obtain all the positive solutions of the equations [Formula: see text] and [Formula: see text] by means of the Fibonacci and Lucas sequences.Furthermore, in this work, we derive some algebraic relations on the Pell form [Formula: see text] including cycle, proper cycle, reduction and proper automorphism of it. We also determine the integer solutions of the Pell equation [Formula: see text] in terms of [Formula: see text] We extend all the results of the papers [3, 10, 27, 37].

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 Diophantine equations z^2 = f(x)^2 ± f(x)f(y) + f(y)^2

2021 ◽  
Vol 27 (2) ◽  
pp. 88-100
Author(s):  
Qiongzhi Tang ◽  
Keyword(s):  
Positive Integer ◽  
Diophantine Equations ◽  
Pell Equation ◽  
Intersection Angle ◽  
Integer Solutions ◽  
Two Sides

Using the theory of Pell equation, we study the non-trivial positive integer solutions of the Diophantine equations $z^2=f(x)^2\pm f(x)f(y)+f(y)^2$ for certain polynomials f(x), which mean to construct integral triangles with two sides given by the values of polynomials f(x) and f(y) with the intersection angle $120^\circ$ or $60^\circ$.

scholarly journals Fibonacci Numbers with a Prescribed Block of Digits

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 639 ◽  
Author(s):  
Pavel Trojovský
Keyword(s):  
Lower Bounds ◽  
Diophantine Approximation ◽  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Algebraic Numbers ◽  
Decimal Expansion ◽  
Linear Forms

In this paper, we prove that F 22 = 17711 is the largest Fibonacci number whose decimal expansion is of the form a b … b c … c . The proof uses lower bounds for linear forms in three logarithms of algebraic numbers and some tools from Diophantine approximation.

The Positive Integer Solutions of a Diophantine Equation

2015 ◽  
Vol 713-715 ◽  
pp. 1483-1486
Author(s):  
Yi Wu ◽  
Zheng Ping Zhang
Keyword(s):  
Positive Integer ◽  
Diophantine Equation ◽  
Pell Equation ◽  
Integer Solutions ◽  
Basic Equations

In this paper, we studied the positive integer solutions of a typical Diophantine equation starting from two basic equations including a Diophantine equation and a Pell equation, and we will prove all the positive integer solutions of the typical Diophantine equation.

scholarly journals A curious property of series involving terms of generalized sequences

2000 ◽  
Vol 23 (1) ◽  
pp. 55-63
Author(s):  
Odoardo Brugia ◽  
Piero Filipponi
Keyword(s):  
Fibonacci Numbers ◽  
Fibonacci Number ◽  
Lucas Numbers ◽  
Generalized Fibonacci Numbers ◽  
Generalized Lucas Numbers ◽  
Curious Property

Here we are concerned with series involving generalized Fibonacci numbersUn  (p,q)and generalized Lucas numbersVn  (p,q). The aim of this paper is to find triples(p,q,r)for which the seriesUn  (p,q)/rnandVn  (p,q)/rn(forrrunning from 0 to infinity) are unconcerned at the introduction of the factorn. The results established in this paper generalize the known fact that the seriesFn/2n(Fnthenth Fibonacci number) and the seriesnFn/2ngive the same result, namely−2/5.

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 The Proof of a Conjecture on the Density of Sets Related to Divisibility Properties of z(n)

Mathematics ◽  
2021 ◽  
Vol 9 (22) ◽  
pp. 2912
Author(s):  
Eva Trojovská ◽  
Venkatachalam Kandasamy
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Fibonacci Sequence ◽  
Natural Density ◽  
Almost All ◽  
Divisibility Properties

Let (Fn)n be the sequence of Fibonacci numbers. The order of appearance (in the Fibonacci sequence) of a positive integer n is defined as z(n)=min{k≥1:n∣Fk}. Very recently, Trojovská and Venkatachalam proved that, for any k≥1, the number z(n) is divisible by 2k, for almost all integers n≥1 (in the sense of natural density). Moreover, they posed a conjecture that implies that the same is true upon replacing 2k by any integer m≥1. In this paper, in particular, we prove this conjecture.

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