k-generalized Fibonacci numbers of the form 1+2^{n_1}+4^{n_2}+\cdots+(2^{k})^{n_k}

Carlos Alexis Gómez Ruiz; Florian Luca

Vol. 19 No. 2 (2014)

Regular Articles

k-generalized Fibonacci numbers of the form 1+2^{n_1}+4^{n_2}+\cdots+(2^{k})^{n_k}

Published 2014-07-06

Carlos Alexis Gómez Ruiz
Florian Luca

Carlos Alexis Gómez Ruiz
Universidad del Valle, Colombia

Florian Luca
UNAM, México and School of Mathematics, University of the Witwatersrand, south Africa

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Abstract

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence (F_n^{(k)})_{n>= 2-k} whose first k terms are 0, ..., 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, we investigate k-generalized Fibonacci numbers written in the form 1+2^{n_1}+4^{n_2}+\cdots+(2^{k})^{n_k}, for non-negative integers n_i, with n_k >= max{ n_i | 1<= i <= k-1}.

Keywords

Fibonacci numbers, Lower bounds for nonzero linear forms in logarithms of algebraic numbers

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Author Biography

Carlos Alexis Gómez Ruiz

Department of Mathematics, Ph.D. student

Florian Luca

Mathematical Institute, Research Professor