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Multiplicative dependence between k-Fibonacci and k-Lucas numbers Carlos A. Gómez1 · Jhonny C. Gómez1 · Florian Luca2,3,4

© Akadémiai Kiadó, Budapest, Hungary 2020

Abstract A generalization of the well-known Fibonacci and Lucas sequences are the k-Fibonacci and k-Lucas sequences with some fixed integer k ≥ 2. For these sequences the first k terms are 0, . . . , 0, 1 and 0, . . . , 0, 2, 1, respectively, and each term afterwards is the sum of the preceding k terms. Here we find all pairs of k-Fibonacci and k-Lucas numbers multiplicatively dependent. Keywords Multiplicatively dependent integers · k-generalized Fibonacci and Lucas numbers · Applications of lower bounds for nonzero linear forms in logarithms of algebraic numbers Mathematics Subject Classification 11B39 · 11D61 · 11J86

1 Introduction For an integer l ≥ 2, we say that the integers A1 , . . . , Al are multiplicatively dependent if there are xi ∈ Z, not all zero, such that A1x1 · · · Alxl = 1. Let F := {Fn }n≥0 be the classical Fibonacci sequence. One important property concerning prime factors of Fibonacci numbers, given by Carmichael’s Primitive Divisor Theorem (see [3]), states that any two distinct Fibonacci numbers with one of the indices greater than

B

Carlos A. Gómez [email protected] Jhonny C. Gómez [email protected] Florian Luca [email protected]

1

Departamento de Matemáticas, Universidad del Valle, Calle 13 No 100-00, Cali 25360, Colombia

2

School of Mathematics, University of the Witwatersrand, Johannesburg, South Africa

3

Research Group in Algebraic Structures and Applications, King Abdulaziz University, Jeddah, Saudi Arabia

4

Centro de Ciencias Matemáticas, UNAM, Morelia, Mexico

123

C. A. Gómez et al.

or equal to 13 are multiplicatively independent. In other words, the Diophantine equation (Fn )x = (Fm ) y , where n > max {12, m} and x, y ∈ Z,

(1.1)

has no solutions (x, y)  = (0, 0). Now, an easy check shows that for indices m < n ≤ 12, the only pairs of multiplicatively dependent Fibonacci numbers correspond to indices (n, m) = (2, 1) and (6, 3). In this paper, for a fixed integer k ≥ 2, we consider a generalization of the Fibonacci (k) sequence, U (k) := {u n }n≥−(k−2) , given by the linear recursion of order k: (k)

(k)

u (k) n = u n−1 + · · · + u n−k for n ≥ 2, (k)

(k)

(k)

(1.2)

(k)

with initial conditions u −(k−2) = · · · = u −1 = 0, u 0 = a, u 1 = b. If a := 0 and b := 1, we obtain the so called k-generalized Fibonacci sequence, F (k) := (k) {Fn }n≥−(k−2) . If instead we take a := 2 and b := 1, we obtain the k-generalized Lucas (k) (k) (k) sequence, L (k) := {L n }n≥−(k−2) . In this context, Fn and L n are known as the nth kFibonacci and k-Lucas numbers, respectively. Despite considerable amount of well known properties between Fibonacci and Lucas num(k) (k) bers, little do we know about multiplicative dependence among Fn and L n . As an example of the problems that we are interested, Gómez and Luca (see [8]) proved the following result related to the Diophantine equatio