MathNet 2007-part-01 — 187 problems (EN–VI) Question 167

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