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

Abstract

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