Question

$f(0)=4$

$f(x+1)=f(x)^2-2$

$\text{if} \quad 2^{x+2}-1|f(x) \quad \text{then} \quad 2^{x+2}-1 \quad \text{is prime}$

How to prove that?

Example:

$x=1$

$f(1)=14$,

$2^{1+2}-1=7$

and

$7|f(1) \implies 7$ is prime.