1
$\begingroup$

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?

  • 2
    What is "the obvious way"? – 2017-01-30
  • 0
    @RobertIsrael, prime generating polynomial is a polynomial that generates prime numbers at some point. For example, $x^2 -x +41$ generates only primes up to $x=40$. – 2017-01-30

2 Answers 2

2

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 $x^3 + 2 y^3$.

1

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 $x$, $0\le x\le44$.

  • 0
    I wrote a code for a cas to find polynomials that consecutively gives primes up to some bound like 44 as you said. I haven't checked whether it can find a polynomial which gives primes for a larger bound because the full program is in my PC at my hometown. I don't know whether a code like mine can be helpful in any ways. Maybe it can help me to find some patterns in some special cases. How can it be useful? – 2017-01-30
  • 0
    I don't know, but that's a whole other question. I'd recommend you do some reading to find out what's been done, and what needs to be done. – 2017-01-30
  • 0
    I am currently reading the chapter you mentioned above and I have a question that may be basic. When we are dealing with a prime producing polynomial $p(x)$, do we check whether $|p(x)|$ or $p(x)$ is prime? – 2017-02-03
  • 0
    I think you get to make up your own rule on that, so long as you make it clear to the reader what definition you are using. Though I think it may be cheating to count both $17$ and $-17$, say, as primes. – 2017-02-03