On the problem of Pillai with k -generalized Fibonacci numbers and powers of 3

Abstract

For an integer k ≥ 2, let {Fn(k)} n≥2-k be the k-generalized Fibonacci sequence which starts with 0,..., 0, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper, we find all integers c with at least two representations as a difference between a k-generalized Fibonacci number and a power of 3. This paper continues the previous work of the first author for the Fibonacci numbers, and for the Tribonacci numbers.