Quadratic forms representing the $p$-th Fibonacci number

In this paper, we show that if $p\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$ is prime, then $4F_p$ admits a representation of the form $u^2-p{v}^2$ for some integers $u$ and $v$, where $F_n$ is the $n$th Fibonacci number. We prove a similar result when $p\equiv -1(\mathrm{m}\mathrm{o}\mathrm{d}4)$.