k-Generalized Lucas numbers, perfect powers and the problem of Pillai

For an integer k >= 2, let L ((k)), let be the k-generalized Lucas sequence which starts with 0,..., 2, 1 (a total of k terms) and for which each term afterwards is the sum of the k preceding terms. In this paper we assume that an integer c can be represented in at least two ways as the difference between a k-generalized Lucas number and a power of b, then using the theory of nonzero linear forms in logarithms of algebraic numbers, we bound all possible solutions on this representation of c in terms of b. Finally, combination our general result and some known reduction procedures based on the continued fraction algorithm, we find all the integers c and their representations for b is an element of [2, 10], this argument can be generalized to any b > 10.