Previous Article in Journal
Self-Similar Models: Relationship between the Diffusion Entropy Analysis, Detrended Fluctuation Analysis and Lévy Models
Previous Article in Special Issue
A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers
Open AccessArticle

# Some Diophantine Problems Related to k-Fibonacci Numbers

by Pavel Trojovský 1,* and Štěpán Hubálovský 2
1
Department of Mathematics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
2
Department of Applied Cybernetics, Faculty of Science, University of Hradec Králové, 500 03 Hradec Králové, Czech Republic
*
Author to whom correspondence should be addressed.
Mathematics 2020, 8(7), 1047; https://doi.org/10.3390/math8071047
Received: 8 June 2020 / Revised: 25 June 2020 / Accepted: 27 June 2020 / Published: 30 June 2020
(This article belongs to the Special Issue New Insights in Algebra, Discrete Mathematics, and Number Theory)
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 $L k , n = L n , k$ . View Full-Text
MDPI and ACS Style

Trojovský, P.; Hubálovský, Š. Some Diophantine Problems Related to k-Fibonacci Numbers. Mathematics 2020, 8, 1047.