Importance of prime generating polynomials

Question

A prime generating polynomial $p(x)$ is defined in the obvious way. It is known that is not possible to find a non-constant polynomial $p(x)$ that generates primes for all integer values for $x$.

I think that the most important question that is related to prime generating polynomials is the Bunyakovsky conjecture . So I wonder whether this area is an active research area? Is it possible to find new important questions related to them?

Answer 1

It would be an important advance if somebody could prove even one nonlinear example of Bunyakovsky's conjecture, i.e. exhibit a polynomial $p(x)$ of degree $> 1$ and prove that it has infinitely many prime values for integers $x$. But I have the impression that this isn't likely any time soon. The closest thing we have to this, AFAIK, are the results of Friedlander and Iwaniec: there are infinitely many primes of the form $x^2 + y^4$ for integers $x,y$; and Heath-Brown: there are infinitely many primes of the form $x^3 + 2 y^3$.

Answer 2

See Chapter 4, Prime-Producing Polynomials, in Ribenboim's book, The New Book of Prime Number Records. He notes, for example, that Ruby found the polynomial $36x^2-810x+2753$, which has distinct prime values for 45 consecutive values of $x$, $0\le x\le44$.