Quadratic forms representing \(p\)th terms of Lucas sequences (Q512239)

Let integers \(r\) and \(s\) be fixed and the Lucas sequence be given by \(A_0 = 0, A_1 = 1, A_n = rA_{n-1} + sA_{n-2}\) for all \(n \geq 0.\) Let \(D = r^2 + 4s \neq 0.\) In this paper, it is proved that: {\parindent=0.7cm\begin{itemize}\item[(1)] If \(p \equiv 1 \pmod 4\) is prime, then \(A_p\) is represented by the quadratic form \(u^2 + uv - \frac{1}{4}(p - 1)v^2\) and \(4A_p\) is represented by the quadratic form \(u^2 - pv^2\). \item[(2)] If \(p \equiv 3 \pmod 4\) is prime, then \(4A_p\) is represented by the quadratic form \(Du^2 + pv^2\). \end{itemize}}