Let $(F_{n})_{n\geq1}$ the Fibonacci sequence. Find all $n \in \mathbb{N}$ such that for every $k=0,1,...,F_{n}$ \[ {F_{n}\choose k} \equiv (-1)^{k} \ (mod \ F_{n}+1) \]
Source: 2018 Olympic Revenge, Problem 1
Tags: number theory
Let $(F_{n})_{n\geq1}$ the Fibonacci sequence. Find all $n \in \mathbb{N}$ such that for every $k=0,1,...,F_{n}$ \[ {F_{n}\choose k} \equiv (-1)^{k} \ (mod \ F_{n}+1) \]