The Distribution of Fibonacci Numbers Modulo Primes

We give a new proof of the Shah–Bruckner theorem, which states that |Ωp|

7, where Ωp is the set of Fibonacci numbers modulo p. We also show that liminfp→∞|Ωp|/p=0; in other words, for every ϵ>0, there exists a prime p such |Ωp|