Lucas sequences and repdigits

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.

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The diophantine equation x2 + paqb = yq

International Journal of Number Theory ◽

10.1142/s1793042121500792 ◽

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.

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Solution of certain Pell equations

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500560 ◽

2018 ◽

Vol 11 (04) ◽

pp. 1850056 ◽

Cited By ~ 1

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].

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An upper bound for the moments of a GCD related to Lucas sequences

Rocky Mountain Journal of Mathematics ◽

10.1216/rmj-2019-49-3-887 ◽

2019 ◽

Vol 49 (3) ◽

pp. 887-902

Author(s):

Daniele Mastrostefano

Keyword(s):

Upper Bound ◽

Lucas Sequences

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On some new results for the generalised Lucas sequences

Analele Universitatii Ovidius Constanta - Seria Matematica ◽

10.2478/auom-2021-0002 ◽

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.

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