Question 100
Let $F_{0}=0$, $F_{1}=1$ and $F_{n+1}=F_{n}+F_{n-1}$, for all positive integer $n$, be the Fibonacci sequence. Prove that for any positive integer $m$ there exist infinitely many positive integers $n$ such that
$$
F_{n}+2 \equiv F_{n+1}+1 \equiv F_{n+2} \quad \bmod m
$$
$$
F_{n}+2 \equiv F_{n+1}+1 \equiv F_{n+2} \quad \bmod m
$$