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Dirichlet's theorem for arithmetic progressions says that there are infinitely many numbers inside every primitive arithmetic progression.

What is known about progressions of the form $P(k)$, with $P$ a polynomial of degree $d$, supposed to be primitive and not vanishing identically on $\mathbb Z$ for every prime number $p$ (such a condition can be effectively tested, since the $p$ for which $P$ vanishes identically mod $p$ are $\leq d$) ?

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    See Bunyakovsky conjecture https://en.wikipedia.org/wiki/Bunyakovsky_conjecture – QC_QAOA Oct 03 '21 at 07:57
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    For polynomials in one variable, no example with degree greater than $1$ is known to produce infinite many primes, not even $n^2+1$. For two variables, surprisingly, $a^2+b^4$ has been shown to produce infinite many primes. – Peter Oct 03 '21 at 08:42

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