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

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 .