# What is importance of the Bunyakovsky conjecture?

Bunuyakovsky conjecture states that:

An irreducible polynomial $f(x)$ of degree two or higher with integer coefficients and property that $\gcd(f(1),f(2),......)=1$ generates for natural arguments infinitely many prime numbers.

Wikipedia article claims that this is an important open problem.

My question is: how important do you consider the answer to this problem, and why?

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It would be nice if those voting to close gave us a hint. –  Gerry Myerson Nov 4 '11 at 11:18
@Gerry, see Too many soft questions?, but perhaps that does not really apply to this question. –  lhf Nov 4 '11 at 11:38
@lhf, thanks. So, closing-voters, is that the reason? The question is too soft? –  Gerry Myerson Nov 4 '11 at 11:44
@Gerry, I did not vote to close. You may check that those who did chose the not constructive enough reason. As you know, this stipulates that This question is not a good fit to our Q&A format. We expect answers to generally involve facts, references, or specific expertise; this question will likely solicit opinion, debate, arguments, polling, or extended discussion. Applying this to the question at hand does not seem completely unfair. –  Did Nov 4 '11 at 11:56
@GerryMyerson Yes, I consider "how important do you consider the answer to this problem" to be too soft (as in "not a good fit to our Q&A format: <...> will likely solicit opinion <...>'"). –  Grigory M Nov 4 '11 at 12:18

As a conjecture it is not important in the same sense as the Weil or Langlands conjectures or the formulation of class field theory, where asking the question properly is a major discovery. It is more of a single name for a broad family of similar problems (such as primes of type $x^2+1$) where congruence obstructions seem to be the only reason a sequence could fail to have infinitely many primes. The conjecture often goes without a name, and I think there are many mathematicians who know the statement of the conjecture but no name for it.
Currently, we know that linear polynomials generate primes, but we know nothing for any non-trivial polynomial of higher degree. We can't prove, for example, that $x^2+1$ is prime for infinitely many $x$. A proof of Bunyakowsky would settle the whole polynomial-prime question all at once, and would be a tremendous advance. So the consequences would be enormous. Whether that makes it important is another question.