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Abstract For an integer k 2 k≥2, let L^ (k) L (k) be the k –generalized Lucas sequence which starts with 0, , 2, 1 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 2, 10 b∈2, 10, this argument can be generalized to any b> 10 b>10.
Faye et al. (Thu,) studied this question.