Powers of two as sums of two k-Fibonacci numbers

Abstract

For an integer k ≥ 2, let (F (k) n )n be the k−Fibonacci sequence which starts with 0, . . . , 0, 1 (k terms) and each term afterwards is the sum of the k preceding terms.In this paper, we search for powers of 2 which are sums of two k−Fibonacci numbers.The main tools used in this work are lower bounds for linear forms in logarithms and a version of the Baker-Davenport reduction method in diophantine approximation. This paper continues and extends the previous work of [3] and [6].

Mersenne k-Fibonacci numbers

“…So, by computing norms from Q(α) to Q, for example, we see that the number f k (α) is not an algebraic integer. Proofs for this fact and (1.5) can be found in [5].…”

Effective results for linear equations in members of two recurrence sequences

“…General results on linear equations involving recurrence sequences have been made most prominently by Schlickewei and Schmidt [17,18]. Recently the case of linear equations involving Fibonacci numbers and powers of two have been picked up by several authors such as Bravo and Luca [6] and Bravo, Gómez and Luca [5] who studied the Diophantine equations F n + F m = 2 a and F denote the sequence of Fibonacci and k-Fibonacci numbers. Besides, Bravo, Faye and Luca [4] studied the Diophantine equation P l + P m + P n = 2 a , where (P n ) ∞ n=0 is the sequence of Pell numbers.…”

Powers of Two as Sums of Three Pell Numbers

“…in integers n ≥ m ≥ ℓ ≥ 0 are in {(2, 1, 1, 2), (3, 2, 1, 3), (5, 2, 1, 5), (6,5,5,7), (1, 1, 0, 1), (2, 2, 0, 2), (2, 0, 0, 1), (1, 0, 0, 0)}.…”