Does the sequence $x_0=8$ , $x_{n+1}=4x_n+1$ contain a prime?
No: $$ 4^n\cdot 8+\frac{4^n-1}{3}=\frac{(5\cdot 2^n+1)(5\cdot 2^n-1)}{3}. $$ (both factors at the numerator are $>3$).
Does the sequence $x_0=8$ , $x_{n+1}=4x_n+1$ contain a prime?
No: $$ 4^n\cdot 8+\frac{4^n-1}{3}=\frac{(5\cdot 2^n+1)(5\cdot 2^n-1)}{3}. $$ (both factors at the numerator are $>3$).