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Inspired by this question, I was wondering if there was a simple proof that if $p(x)$ is a polynomial of degree $\geq 1$ and with integer coefficients, then $p(x)$ is a composite number for infinitely many integers $x.$

We can split it into reducible and irreducible polynomials over either $\mathbb{Z}$ or $\mathbb{R}$. But I don't see how that helps actually.

This paper seems relevant, but to me it is not so simple to understand (for example I am not super familiar with Gauss' Lemma) and I wonder if a simpler solution exists. Maybe the fact that I am not familiar with Gauss' Lemma means I should not be asking this question? If this is the feedback, then so be it.

Adam Rubinson
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  • Almost surely this is the case , otherwise we would have a polynomial $p(n)$ being prime for EVERY sufficient large integer $n$ , extremely hard to imagine. But it is probybla nevertheless an open problem. – Peter Jul 07 '23 at 17:35

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