# There are infinitely prime numbers $p$, st $P(x)+p$ is irreducible over $\mathbb{Z}[x]$

According to the Bunyakovsky conjecture it's an open problem if under special conditions, integer polynomials of degree greater than one generate infinitely many primes.

Does anyone know if the following problem, involving integer polynomials and prime numbers, has ever been studied?

Consider a polynomial $P(x) \in \mathbb{Z}[x]$ of degree $\geq 2$. Are there infinitely many prime numbers $p$ such that $P(x)+p$ is irreducible over $\mathbb{Z}[x]$?

Any help would be appreciated.

• This should be the case, but I have no idea for a proof yet. Mar 17, 2018 at 11:46

## 1 Answer

Yes by the Hilbert Irreducibility Theorem. In Serre's "Topics in Galois Theory" is explained that (1) $P(x)+q$ is irreducible over $\mathbb{Q}$ except for a "thin" subset of $q \in \mathbb{Q}$ and (2) the number of integer points of height $\leq N$ in a thin subset of $\mathbb{Q}$ is $O(N^{1/2})$; thus the affirmative answer follows from the prime number theorem (the number of primes of height $\leq N$ is asymptotic to $N/log(N)$).