Prime values of integer polynomials

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

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