Square-Classes in Lucas Sequences Having Odd Parameters Original Research Article

Two or more terms of a sequence are said to be in the same square-class if the squarefree parts of the terms are identical. Let {Un(P, Q)} and {Vn(P, Q)} denote the Lucas sequence and companion Lucas sequence, respectively, with parametersPandQ. For all odd relatively prime values ofPandQwith discriminantP2−4Q>0, we show that {Un(P, Q)} and {Vn(P, Q)} have only finitely many non-trivial square-classes and each square-class contains at most three terms. The square-classes are explicitly determined in “most” case, and an effectively computable bound on the number of square-classes, depending onPandQ, is obtained in the remaining cases.