Powers of two as sums of two k-Fibonacci numbers

For an integer k ≥ 2, let (Fn(k))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 [4] and [2].