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In this paper, we consider the Diophantine equation Vn-bm=c for given integers b, c with b≥2, whereas Vn varies among Lucas-Lehmer sequences of the second kind. We prove under some technical conditions that if the considered equation has at least three solutions (n, m) , then there is an upper bound on the size of the solutions as well as on the size of the coefficients in the characteristic polynomial of Vn.
Heintze et al. (Mon,) studied this question.