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 . A list of generalized Fermat primes sorted by base $b$ can be found here . I have tested this claim for $b \in [2,10000]$ with $n \in [2,10]$ and there were no counterexamples .