The terms of the form \(7kx^{2}\) in the generalized Lucas sequence with parameters \(P\) and \(Q\) (Q2804253)

The investigation of powers and multiples of powers in second-order linear recurrence sequences has a rich history. In the paper under review, after a detailed description of the known results, the author studies the equation \(V_n=7kx^2\) where \((V_n)\) is the (usual) generalized Lucas sequence with parameters \(P\), \(Q\) and \(k>1\) is an integer. He shows that whenever the equation has a solution, then \(n=1,3,5\) provided \(k\) divides \(P\) and \(P, Q\) are odd and relatively prime. The method of proof is similar to the one used by Cohn, McDaniel and/or Ribenboim, namely, a careful inspection of the associated Jacobi symbols and congruences.