On the {L}ucas sequence equations {\(V_n(P,1)=wkx^2\)}, {\(w\in\{5,7\}\)} (Q1677556)

In this paper, the author considers the Lucas sequence of the second kind \(\{V_n\}_{n\geq 0}\) given by \(V_0=2,~V_1=P,~V_{n+2}=PV_n+V_{n-1}\), where \(P\) is an odd positive integer and solves the equations \(V_n=5k x^2,~7kx^2,~5kx^2\pm 1\), where \(k\) is any divisor of \(P\). For example, the first main result asserts all solutions to the first of above equations \(V_n=5kx^2\) have \(n=1,3\). He applies these results to prove that if \(V_n=Ax^2\) and \(A\in \{15,21,35\}\), then \(n=1,3\). Except for one equation for which the author used MAGMA to find all the integral points on a certain elliptic curve, all the other equations are dealt with by elementary but clever manipulations with Jacobi symbols.