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 $F_n + 2 \equiv F_{n+1} + 1 \equiv F_{n+2}$ mod $m$ .