Square-classes in Lucas sequences having odd parameters (Q1281629)

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 \(\{U_n (P,Q)\}\) and \(\{V_n(P,Q)\}\) denote the Lucas sequence and companion Lucas sequence, respectively, with parameters \(P\) and \(Q\). For all relatively prime values of \(P\) and \(Q\) with discriminant \(P^2-4Q>0\), it is shown that \(\{U_n(P,Q) \}\) and \(\{V_n(P,Q)\}\) have only finitely many nontrivial square-classes and each square-class contains at most three terms. The square-classes are explicitly determined in ``most'' cases, and an effectively computable bound on the number of square-classes, depending on \(P\) and \(Q\), is obtained in the remaining cases.