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

For an integer \(k\ge 2\) $k\ge 2$ , let \(L^{(k)}\) $L^{(k)}$ be the k–generalized Lucas sequence which starts with \(0, \ldots , 2,1\) $0,\dots ,2,1$ (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 \( b\in [2,10]\) $b\in [2,10]$ , this argument can be generalized to any \( b> 10 \) $b>10$ .