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?



Integer dynamics hitting infinitely many primes
I'm fairly certain that nothing along these lines is known.


Until the late 1990's there were no known `elementary' polynomial sparse sequences that contained infinitely many primes(\*). In 1998 Friedlander and Iwaniec proved that the sequence of integers of the form $x^2 +y^4$ contains infinity many primes. The number of elements of this sequence less than $N$ is $N^{3/4}$. In 2001, Heath-Brown proved that the sequence of integers of the form $x^3+2y^3$ contain infinitely many primes. The number of elements of this sequence less than $N$ is $N^{2/3}$. This seems the best result in this direction to date. Nothing seems to be known (other than upper bounds from sieve theory) about polynomials in a single variable (other than linear equations / arithmetic progressions, of course).


If there was a polynomial whose iterates contained infinity many primes this would give an exponentially sparse sequence with infinitely many primes. This seems far from anything anyone can prove at present. In particular, there appears to be no polynomial time (in the bit length) constructable sequence of integers which is proven to be prime infinitely often and whose intersection with the first $N$ integers is less than $N^{1/2}$. Indeed such a result would yield new results on the problem of deterministically constructing large primes (see the [Polymath 4 project](http://arxiv.org/abs/1009.3956)).


On the other hand, there is some remarkable work in which nontrivial estimates have been obtained for the number of prime divisors of sequences obtained by iterates of various quadratic polynomials. See: R. Jones, [The density of prime divisors in the arithmetic dynamics of quadratic polynomials](http://www.ams.org/mathscinet-getitem?mr=2439638). J. Lond. Math. Soc. (2) 78 (2008), no. 2, 523–544


**Update:** (\*) Indeed, I had forgotten about the [Piatetski-Shapiro prime number theorem](http://www.ams.org/mathscinet-getitem?mr=59302) when I wrote this. However, the best exponent on this result is gives a set of density around $N^{.861}$ which is inferior to Heath-Brown's result for the purposes discussed here.


