Reference request for a divisibility property of Fibonacci numbers Define the Fibonacci numbers $F_n$ by $F_n=F_{n-1}+F_{n-2}$ and initial values $F_0=0$ and $F_1=1.$
I would like to get a reference for the following result:
If $p$ is a prime number with  $p \equiv  \pm 2\bmod 5$ then $p$ divides $F_{p+1}$.

On this topic, maybe someone knows of an easy way to use Hardy and Wright to show this result that a student of mine has discovered independently on her own: for $p$ prime, $F_p \equiv 1$ mod $p$ if $p \equiv 1,4$ mod 5, and $F_p \equiv -1$ mod $p$ if $p \equiv 2,3$ mod 5. In other words, $F_p \equiv \left(\frac{p}{5}\right)$ for $\left(\frac{p}{5}\right)$ the Jacobi symbol.
I can't quite see how to get there from Hardy and Wright's Theorem 180.
