Treffer: The Square Terms in Lucas Sequences: The square terms in Lucas sequences

Let \(P\) and \(Q\) be relatively prime odd integers and define the sequences \(\{U_n\}\) and \(\{V_n\}\) by \(U_n = PU_{n - 1} - QU_{n - 2}\) with \(U_0 = 0\), \(U_1 = 1\) and \(V_n = PV_{n - 1} - QV_{n - 2}\) with \(V_0 = 2\), \(V_1 = P\). The main results of the paper are the following. (i) If \(V_n\) is a square, then \(n = 1,3\) or 5. (ii) If \(2V_n\) is a square, then \(n = 0,3\) or 6. (iii) If \(U_n\) is a square, then \(n = 0,1,2,3,6\) or 12. (iv) If \(2U_n\) is a square, then \(n = 0,3\) or 6. These results are nice extensions of some earlier ones concerning special (e.g. Fibonacci and Lucas) sequences.