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.

 

[blue_raver22]

Hello there! This is an interesting question and one that has been studied by mathematicians for quite some time. The conjecture you mentioned is known as the Schinzel's hypothesis H and it states that every monic irreducible polynomial with integer coefficients has infinitely many prime values. This problem has been open for many years and has been studied by many mathematicians, but it remains unsolved.

Some progress has been made in special cases, such as when the polynomial has a certain form or when it has a specific degree. However, a general proof for all monic irreducible polynomials is still elusive. In fact, proving this conjecture would have major implications in number theory and algebraic geometry.

There are also some related conjectures, such as the Bateman-Horn conjecture, which states that there are infinitely many primes of the form f(n), where f is a fixed polynomial with integer coefficients. This conjecture has been proven for certain families of polynomials, but again, a general proof is still unknown.

In summary, the problem you mentioned is a well-known and important open problem in mathematics. While progress has been made, it remains unsolved and is an active area of research. I hope this helps answer your question!