Reference request for a divisibility property of Fibonacci numbers

Question

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 $F_{p+1}$.

Answer 1

In Fibonacci Numbers Modulo $p$ by Brian Lawrence (Theorem 4.4), or in (1.5) from Some formulae for the Fibonacci sequence with generalizations by George H. Andrews. The latter cites G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 1960, page 150. Here is a screen copy (and a copy of the book):

Answer 2

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.