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?