Let $P$ be a polynomial having integer coefficients (and degree $\geq 3$), and let $\mathscr P\_P$ be the set of prime numbers dividing some value $P(n)$ with $n \in \mathbb Z$.



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> Is it true that $\sum\_{p \in \mathscr P\_P} \frac1 p$ diverges?
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The case of polynomials with degree $2$ can be studied by Dirichlet's theorem on arithmetic progressions (I suppose a quadratic-residue argument can be applied for every case), but I am not sure how to study the cases of greater degree. 


Any reference or proof (*or* disproof, which, by the way, would be very surprising) would be appreciated.



About the prime divisors of values of polynomials
We will assume that $f$ is irreducible over $\mathbb{Q}$. Let $\gamma$ be a root of $f$, and let $K = \mathbb{Q}(\gamma)$. By Chebotarv's density theorem, the set $P\_f$ of primes $p$ with the property that $f(x) \equiv 0 \pmod{p}$ has at least one root in $\mathbb{F}\_p$ has positive relative density in the set of rational primes, and the set of integers with at least one prime factor in $P\_f$ has positive density. For each prime in $P\_f$, we can solve the congruence $f(x) \equiv 0 \pmod{p}$ in the integers; hence $P\_f$ is precisely the set of primes you are looking for. Since $P\_f$ has positive relative density in the primes, the sum of their reciprocals will diverge, with the constant as predicted by Chebotarv's density theorem.


