Can you provide a proof or a counterexample for the following claim :

>Let $P_m(x)=2^{-m}\cdot((x-\sqrt{x^2-4})^m+(x+\sqrt{x^2-4})^m)$ . 
Let $F_n(b)= b^{2^n}+1 $ where $b$ is an even integer , $ 3\nmid b , 5\nmid b  $ and $n\ge2$ . Let $S_i=P_b(S_{i-1})$ with $S_0=P_{b/2}(P_{b/2}(8))$ , then $F_n(b)$ is prime iff $S_{2^n-2} \equiv 0 \pmod{F_n(b)}$ .

You can run this test [here][1] . 


 


  [1]: https://sagecell.sagemath.org/?z=eJxtjD0OwjAMhXckLtEppkElFgNS8NqNK3RICEqkNonqCMTtcQcWxOLn96PPEV7sfpcJ5TLhoZbZx-DeHMNTuQG10X-iM4DsR3ITTrk38vu20qavmOagxF1JqiNqplu5q1-wMHhA0CNY2fZkNl56KCY66bqm3FQnsoQOvtaXpRZOTSKwH172Nto=&lang=gp