Abstract
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 .
Highlights
Let ( Fn )n be the Fibonacci sequence given by Fn+2 = Fn+1 + Fn, for n ≥ 0, where F0 = 0 and
The ratio of two consecutive of these numbers converges to the Golden section α = (1 + 5)/2
By using Binet’s formula of k-Fibonacci numbers, it is a simple matter to prove that lim n→∞
Summary
The many identities related to Fibonacci numbers, we cite. Marques and Togbé [5] searched for similar identities in higher powers. They proved that if Fns + Fns +1 is a Fibonacci number for all sufficiently large n, s = 1 or s = 2. We shall work on some Diophantine problems related to these sequences. Our first result concerns the search for higher power identities related to (2) and (3) in the spirit of the Marques and Togbé paper ([5]). Several problems in number theory are questions about the intersection of two known sequences (or sets). For Theorems 3–5, we shall use a plenty of inequalities together with some (combinatorial) identities for Fk,n and Lk,n
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