TITLE: Integer dynamics hitting infinitely many primes
QUESTION [11 upvotes]: I am wondering if there are any rigorous results telling that some dynamical system hits infinitely many primes (except for the case when orbits are just arithmetic progressions). To make it specific, is there a polynomial $f(x)$ such that its iterations $f^n(x)$ are prime for infinitely many $n$ given that the integer $x$ is fixed. A simple observation here is that if $f(x) = 2x + 1$ then asking if iterations of $f(1)$ contain infinitely many primes is equivalent to the Mersenne primes conjecture. 
P. S.
This question seems to be related to the Bunyakovsky conjecture, so maybe somebody knows about any partial results in this direction?

REPLY [7 votes]: In 1947, American William . H. Mills proved that there is a real number $A$, greater than 1 but not an integer, such that integer part of
$$
A^{3^n} 
$$
is prime for all $n =1, 2, 3, \ldots$. The numnber $A$ is known as the Mill's constant. Its value is unknown, but if the Riemann hypothesis is true then,
$$
A \approx 1.3063778838630806904686144926\ldots
$$
[1] Mills, W. H. (1947) A prime representing function. Bull. Amer. Math. Soc., 53: 604: MR 8, 567.
[2] [Mill's constant]1