On Lucas sequence terms of the form \(kx^2\) (Q2710132)

Let \(P\) and \(Q\) be coprime odd integers with \(P^2-4Q>0\), and let \(U_n=(\alpha ^n-\beta ^n)/(\alpha -\beta)\) and \(V_n=\alpha ^n+\beta ^n\) be the usual linear recursive sequences associated to the polynomial \(X^2-PX+Q=(X-\alpha)(X-\beta)\). For a square-free integer \(k\), define condition (H) for all integers \(u\geq 1\) and for each divisor \(h\) of \(k\), the Jacobi symbol \((-V_{2^u} |h)\) is defined and equals \(+1\). One of the main results of this paper is the following NEWLINENEWLINENEWLINETheorem: Let \(n\geq 0\). If (H) holds and \(U_n=kx^2\) then \(n=0\), \(1\), \(2\), \(3\), \(6\) or \(12\). NEWLINENEWLINENEWLINEWhen \(k\) divides \(P^2-Q\), the author gives necessary and sufficient conditions such that \(U_n=kx^2\), and treats the case \(k=3\). The last section contains results on the equation \(V_n=kx^2\). The proofs use only elementary arguments.NEWLINENEWLINEFor the entire collection see [Zbl 0959.00053].