What is importance of the Bunyakovsky conjecture?

Question

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?

Answer 1

As a matter of empirical science (i.e., concerning only whether the statement is likely to be true, not proofs of the conjecture) there is an important question of whether the methods for predicting the asymptotic number of primes in sequences are reliable. Polynomial sequences are not generic but are of clear interest as a test case where there is algebraic structure. Knowing to what extent probabilistic models of the prime number distribution are correct, and how they are related to algebraic geometry, is a basic motivating question in number theory.

As a part of mathematical theory, whatever the methods are that could prove the predictions, having them would be a huge advance that almost certainly would lead to many other good things. For instance, a technique for proving quasirandom properties that is strong enough to apply to primes, or a new level of understanding in analytic number theory, or a usable theory of prime solutions of algebraic equation similar to the Diophantine geometry of integer or rational points.

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.

Answer 2

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.