k - Fibonacci numbers close to a power of 2

Abstract

A generalization of the well-known Fibonacci sequence is the k-generalized Fibonacci sequence F^(ƙ) := (F_n^(ƙ) )_{n ≥ 2 - ƙ} whose first ƙ terms are 0,..., 0,1 and each term afterwards is the sum of the preceding ƙ 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^(ƙ) 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.