Prime values of integer polynomials

[VKint]

Hey there, physics forums!

A question occurred to me the other day: Is it true that if $$f \in \mathbb{Z}[x]$$ is monic and irreducible over $$\mathbb{Q}$$, then for at least one $$a \in \mathbb{Z}$$, $$f(a)$$ is prime? I can't prove it, but I suspect it's true. Does anyone know if this problem has been solved?

 

[Petek]

What about f(x) = x^2 + x + 4? That's monic and irreducible over Q but isn't prime for any x in Z.

 

[VKint]

Good point.

 

[CRGreathouse]

The Hardy-Littlewood conjectures discuss this in detail, at least for quadratic polynomials (and one cubic). I can't remember who first published the analysis of higher-order polynomials, but you could probably find it on Google.