I have a conjecture:
For any integer $N$, there exist an positive integer $n > N$ such that there exist a degree-$n$ polynomial $P(x)$ satisfying:the sequence $\left\{P(n) \right\}$ contains infinitely many primes.
The conjecture came to me when I was trying to solve a problem concerning irreducible polynomials and prime numbers. This seems reasonable since we have a known result for degree-$1$ polynomials(Dirichlet's theorem). Furthermore, we can show that the sequence $\left\{n^2 + 1 \right\}$ contains infinitely many square-free integers by easy sift-method.
Thanks in advance.
Best Answer
That conjecture is a consequence of "Schinzel's hypothesis H". See here.
In particular, Schinzel's hypothesis H implies that any irreducible polynomial $f$ for which $f(n)$ is not always divisible by a fixed positive integer, is prime infinitely often.
The condition "not always divisible" takes care of $f(x)=x^N + x + 2$, for which $f(n)$ is always even.
It is possible I misunderstood the question because it was unclear what quantifiers you intended on $n$ and $P$. If $P=P_n$ is supposed to have degree $n$, then what you wrote should still follow from standard conjectures, but perhaps not as trivially.