If $p$ is prime, then $F_{p-\left(\frac{p}{5}\right)}\equiv 0\bmod p$, where $F_j$ is the $j$th Fibonacci number, and $\left(\frac{p}{5}\right)$ is the Jacobi symbol.
Who first proved this? Is there a proof simple enough for an undergraduate number theory course? (We will get to quadratic reciprocity by the end of the term.)
Best Answer
I recommend Chapter XVII of Volume 1 of Dickson's History of the Theory of Numbers. He cites results of Legendre, Gauss, Dirichlet, and Lagrange, among others, none of them exactly the one you cite, but all of them very closely related, and more general.
Hardy and Wright give two proofs. It's Theorem 180 in the chapter on continued fractions, and then there's another proof at the end of section 4 of Chapter 15, Quadratic Fields.
See also Theorem 4.12 of Niven, Zuckerman, and Montgomery.