On a problem of Pillai with k–generalized Fibonacci numbers and powers of 2

For an integer k ≥ 2, let {F n } n≥0 be the k–generalized 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 find all integers c having at least two representations as a difference between a k–generalized Fibonacci number and a power of 2 for any fixed k ≥ 4. This paper extends previous work from Ddamulira et al. (Proc Math Sci 127(3): 411–421, 2017. https://doi.org/10.100/s12044-017-0338-3) for the case k = 2 and Bravo et al. (Bull Korean Math Soc 54(3): 069–1080, 2017. https://doi.org/10.4134/BKMS.b160486) for the case k =3.