<i>k</i> −Fibonacci numbers close to a power of 2

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.