I just wondered about the following question:
Suppose that we are given a homogeneous second-order recurrence relation, $x_{n+2}+ax_{n+1}+bx_n=0$ for all $n\in\mathbb{N}$.
Can we choose integers $a$, $b$, $x_0$ and $x_1$ in such a way that the resulting sequence $(x_n)$ will contain infinitely many primes?
Does anybody know anything about this? Or any ideas? I couldn't come up with much, having done some years of maths at uni.