Fibonacci and k Lucas Sequences as Series of Fractions

2016 ◽  
Vol 4 (2) ◽  
pp. 107-119
Author(s):  
A. D. Godase ◽  
M. B. Dhakne
Keyword(s):  
Lucas Sequences

scholarly journals Elliptic Solutions of Dynamical Lucas Sequences

Entropy ◽  
2021 ◽  
Vol 23 (2) ◽  
pp. 183
Author(s):  
Michael J. Schlosser ◽  
Meesue Yoo
Keyword(s):  
Discrete Time ◽  
Time Dependent ◽  
Fibonacci Polynomials ◽  
Lucas Sequences ◽  
Elliptic Solutions

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler–Cassini identity.

The diophantine equation x2 + paqb = yq

2021 ◽  
pp. 1-18
Author(s):  
Hemar Godinho ◽  
Victor G. L. Neumann
Keyword(s):  
Diophantine Equation ◽  
Prime Numbers ◽  
Natural Numbers ◽  
Lucas Sequences ◽  
Primitive Divisors

In this paper, we consider the Diophantine equation in the title, where [Formula: see text] are distinct odd prime numbers and [Formula: see text] are natural numbers. We present many results given conditions for the existence of integers solutions for this equation, according to the values of [Formula: see text] and [Formula: see text]. Our methods are elementary in nature and are based upon the study of the primitive divisors of certain Lucas sequences as well as the factorization of certain polynomials.

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].

scholarly journals On some new results for the generalised Lucas sequences

2021 ◽  
Vol 29 (1) ◽  
pp. 17-36
Author(s):  
Dorin Andrica ◽  
Ovidiu Bagdasar ◽  
George Cătălin Ţurcaş
Keyword(s):  
Integer Sequences ◽  
Lucas Sequences ◽  
Lucas Sequence ◽  
Necessary And Sufficient

Abstract In this paper we introduce the functions which count the number of generalized Lucas and Pell-Lucas sequence terms not exceeding a given value x and, under certain conditions, we derive exact formulae (Theorems 3 and 4) and establish asymptotic limits for them (Theorem 6). We formulate necessary and sufficient arithmetic conditions which can identify the terms of a-Fibonacci and a-Lucas sequences. Finally, using a deep theorem of Siegel, we show that the aforementioned sequences contain only finitely many perfect powers. During the process we also discover some novel integer sequences.

scholarly journals Practical numbers in Lucas sequences

2018 ◽  
Vol 42 (7) ◽  
pp. 977-983 ◽  
Author(s):  
Carlo Sanna
Keyword(s):  
Lucas Sequences
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