CAPS 2007 Problem 2

The Fibonacci sequence is defined by $a_1=a_2=1$ and $a_{k+2}=a_{k+1}+a_k$ for $k\in\mathbb N.$ Prove that for any natural number $m,$ there exists an index $k$ such that $a_k^4-a_k-2$ is divisible by $m.$