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There's this theorem by T. Nagell which states:

Given two non-constant polynomials $P,Q \in \mathbb{Z}[X]$, there exists infinitely many primes $p$ which divide $P(a), Q(b)$ for some integers $a,b$.

I've been trying to find a proof online but was not able to. I'm wondering if anyone has a (preferably elementary) proof of this result.

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    $\begingroup$ Related answer: math.stackexchange.com/a/4265051/1009203 $\endgroup$ Commented Nov 21 at 18:49
  • $\begingroup$ Let $K$ be a finite extension of $\mathbb{Q}$ where $P$ and $Q$ split, there is an infinite number of primes $p$ that split completely in $K$ by Chebotarev theorem. For such primes $p$, the Frobenius on the corresponding residual field is trivial so $\overline{P}$ and $\overline{Q}$ split in $\mathbb{F}_p$. There is a more elementary answer using the resultant to reduce to Schur theorem with only one polynomial. $\endgroup$ Commented Nov 22 at 0:52

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