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

For an integer k = 2, let {F ( k) 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/ 10.1007/ 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.