MEMO 2012 I4

The sequence $\{ a_n \} _ { n \ge 0 }$ is defined by $a_0 = 2 , a_1 = 4$ and $$a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1}$$ for all positive integers $n$ . Determine all prime numbers $p$ for which there exists a positive integer $m$ such that $p$ divides the number $a_m - 1$ .