The sequence
![\{ a_n \} _ { n \ge 0 }](/media/m/6/0/d/60d2ef466a39e4d17d75d120b1a60108.png)
is defined by
![a_0 = 2 , a_1 = 4](/media/m/a/5/7/a5733e1e88e606647239479f471bc7a7.png)
and
![a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1}](/media/m/2/8/c/28c4b16ad29e5d5745a6c738e66af1dc.png)
for all positive integers
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
. Determine all prime numbers
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
for which there exists a positive integer
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
such that
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
divides the number
![a_m - 1](/media/m/c/4/6/c46e5ea7eaecbc25d0eed5b2df04b624.png)
.
%V0
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 $a_m - 1$.