K−Fibonacci numbers close to a power of 2

Abstract

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence whose first k terms are 0, . , 0, 1 and each term afterwards is the sum of the preceding k terms. In this paper, by using a lower bound to linear forms in logarithms of algebraic numbers due to Matveev and some argument of the theory of continued fractions, we find all the members of F (k) which are close to a power of 2. This paper continues and extends the previous work of Chern and Cui which investigated the Fibonacci numbers close to a power of 2.