The general formula for the Fibonacci numbers is
Fn = (1 /√5 )(αn − βn), α = (1 + √5)/2 , β = (1 − √5)/2
If p is a prime such that p ≡ 1, 4 (mod 5), prove that the period of Fn (mod p) divides p − 1. In particular, Fp ≡ 1 (mod p). [Hint: interpret α, β as elements of Fp.]