TITLE: Polynomials that pass through a lot of primes
QUESTION [12 upvotes]: Whenever I mention a polynomial $p(x)$ passes through $n$, I mean there is an integer $z$ so that $p(z)=n$, also any polynomial I talk about is assumed to have integer coefficients.
It's a well known open to question to find a polynomial of degree 2 or more that passes through an infinite amount of primes.
Do we know of general polynomials of degree n that pass through at least f(n) primes where f isn't constant (so f could be like log(n) or 1.5*n)?
Additionally, do we know if the number of primes a quadratic pass through is unbounded (and similarly for other degrees)? Meaning if we look at $s(p(x))$ : the number of primes $p(x)$ passes through, where $p(x)$ is a quadratic, is $s(p(x))$ bounded?

REPLY [3 votes]: Your question is very difficult to answer in full.
For the quadratic case there are partial answers. I have kept, among others and for some time now, an article of Betty Garrison (San Diego State University. A.M.M. 97 (1990) p. 316-17) in which there is the following improvement of Sierpinski's result:
THEOREM. Let $k\ge 2$  be an integer and let $ M $ an arbitrarily large number. Then there exist positive integers $c,d$ such that  $x^k+c$, likewise $x^k-d$ is prime for more than $M$ positive integers $x$
The famous conjecture of Buniakowski (1854) say that for any irreducible polynomial $ f(x)\in \mathbb  Z [x]$ such that the set of values f (n) has no common divisor larger than $1$ ($f(x)=x^3+x+2$ is irreducible but $f(n)$ is always even) there are infinitely many primes $f(n)$. This conjecture is still one of the major unsolved problems in number theory when the degree of $f$ is greater than one. For degree 1 one has  Dirichlet’s theorem on primes in arithmetic progressions (obviously $ax+b; (a,b)=1$, is irreducible);  in other words, Buniakowski’s conjecture wants to generalize this celebrated theorem.