TITLE: Generating primes via composition of polynomials
QUESTION [5 upvotes]: Is it possible that there is a polynomial $f$ with integer coefficients and an integer $a$ such that the sequence $f(a), f(f(a)), \ldots$ consists only of primes and tends to infinity? Are there any specific polynomials $f$ that are conjectured to have this property?

REPLY [2 votes]: It is not known whether there exists a univariate polynomial with integer coefficients of degree at least $2$ that assumes an infinite number of values that are prime. See the Bunyakovsky conjecture in this context -  http://en.wikipedia.org/wiki/Bunyakovsky_conjecture. 
The polynomial $f$ with $f^n(a)$ prime would produce infinitely many primes as values. I would believe that $f$ must be constant, but I don't know how difficult it is to prove this. Some of these questions are very hard, as mentioned before.