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A Quadratic Diophantine Equation Involving Generalized Fibonacci Numbers
Open AccessArticle

Some Diophantine Problems Related to k-Fibonacci Numbers

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
Keywords: k-Fibonacci number; k-Lucas number; Galois theory; Diophantine equation k-Fibonacci number; k-Lucas number; Galois theory; Diophantine equation
MDPI and ACS Style

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

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