two questions about primes

Question

I'm very ignorant about results in number theory concerning the primes. Please let me know if these are open conjectures or easy problems:

1. There are infinitely many primes of the form $n!+1$

2. There is a polynomial with integer coefficients which is prime for infinitely many integer inputs.

Edit: Yeah sorry I meant when the degree is at least 2. I know of Dirichlet for degree 1.

Answer 1

1. These (and those of the form $n!-1$) are called factorial primes. We don't know if there are infinitely many.

2. As pointed out by others, the polynomial $x$ works. More generally, $ax + b$ works when $a$ and $b$ are coprime; this follows from Dirichlet's theorem. For polynomials of degree at least two, not much is known. Even $x^2 + 1$ is a mystery. Iwaniec proved that there are infinitely many $n$ for which $n^2 + 1$ has at most $2$ prime factors. If a polynomial map $f$ takes on infinitely many prime values, then $f$'s leading coefficient is positive, $f$ is irreducible over the integers, and the numbers $f(1)$, $f(2)$, $f(3)$, $\dotsc$ do not have a common factor bigger than one. Bunyakovsky's conjecture asserts that these necessary conditions are in fact sufficient. (To check the last condition, it is enough to find just one pair $n$, $m$ of integers with $f(n)$, $f(m)$ coprime.) We don't know any degree-at-least-two polynomial for which the Bunyakovsky conjecture holds.