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Take a polynomial with integer coefficients $p(x)$. Some of them, like $p(x) = x^2$ or $p(x) = x^3-x$ produce very few prime numbers, while some others $p(x) = x^2-x+41$ seem to produce lots of them.

If you start from a simple example $p(x) = ax+b$ with $a,b$ coprime, since primes are evenly distributed among mod a classes, we will have that the probability of $p(x)$ being prime is the same as the probability of $x$ being prime. Do you think this is true for generic polynomials, or there is some intrinsic bias in increasing the degree toward being more composite/ more prime? For me that I am not expert, this seems an incredibly hard question. I don't want necessarily to be quantitative, but for example a weak formulation could be

$$ \lim_{n \to \infty } \frac{ \log |\{x\le n: p(x) \text{ is prime } \}|}{\log(n) } = 1 $$

Andrea Marino
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    I think you want to search for Bateman-Horn. – Gerry Myerson Dec 26 '20 at 02:03
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    This is an incredibly hard question. :-) Even the question of 'is $p(n)$ prime infinitely often?' for irreducible polynomials $p$ of degree $\gt 1$ is wide-open AFAIK; for instance take $p(n)=n^2+1$... – Steven Stadnicki Dec 26 '20 at 02:03
  • @Steven very hard to prove anything, yes, but there are widely-believed conjectures. – Gerry Myerson Dec 26 '20 at 02:04
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    @GerryMyerson this refers to a question in the last few minutes that Andrea answered; most likely the polynomial given there was not correctly transcribed, as it is irreducible and represents many primes. I'll find the link. https://math.stackexchange.com/questions/3961922/find-all-positive-integers-n-such-thatn6-6n5-4n4-n2-n-2-is-pr/3961949#3961949 – Will Jagy Dec 26 '20 at 02:06
  • ok, so you are saying: we dont' know if they are infinite, you are asking for density? Take a seat, please. – Andrea Marino Dec 26 '20 at 02:06
  • @AndreaMarino to whom are you addressing your comment? – Will Jagy Dec 26 '20 at 02:13
  • To steven stadnicki! – Andrea Marino Dec 26 '20 at 02:28
  • As Gerry mentions, this entire circle of questions is a special case of the Bateman-Horn conjecture, which allows for multiple polynomials being prime at the same time, not just a single polynomial, e.g., can $n^2 + 1$ and $n^3 - 2$ both be prime together for infinitely many $n$? All such questions are completely out of reach except for a single polynomial of degree $1$ (Dirichlet's theorem). – KCd Dec 26 '20 at 02:30
  • You bring up the example $x^2 - x + 41$, which has the peculiar feature of taking lots of initial prime values at small integers. This is a different issue from how often the polynomial takes prime values over long intervals of integers. That "many initial values are prime" property is related to class numbers of quadratic fields: $x^2 - x + 41$ has discriminant $1 - 4(41) = -163$ and $\mathbf Q(\sqrt{-163})$ is the last imaginary quadratic field with class number $1$. See https://math.stackexchange.com/questions/2561/eulers-remarkable-prime-producing-polynomial-and-quadratic-ufds. – KCd Dec 26 '20 at 02:33
  • Yes, you are right, that's a different issue and it was just folklore: the Bateman-Horn conjecture seems exactly what I was looking for. I am very excited to try to apply this to the OP question here: https://math.stackexchange.com/questions/3961922/find-all-positive-integers-n-such-thatn6-6n5-4n4-n2-n-2-is-pr/3961949?noredirect=1#comment8170903_3961949. I am pleased by the degree factor in the denominator of the BH conjecture, but I was wondering if it is possible to estimate the numerator even not sharply, but significantly. – Andrea Marino Dec 26 '20 at 02:41
  • By "numerator" do you mean the infinite product over primes? Some discussions about that product are on Mathoverflow: https://mathoverflow.net/questions/214873/bateman-horn-conjecture-continued – KCd Dec 26 '20 at 02:44
  • yep! oh-oh, seems like i should get back to study algebraic number theory to understand this stuff. It's a pity I have never really get into this. However, I imagine that the answer is "no, there is no obvious estimate of the product, but there are clever ones", right? Thank you :) – Andrea Marino Dec 26 '20 at 02:50
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    @AndreaMarino you should begin a comment with an @ sign followed by the username of the person you are addressing. You will be offered a little window with a choice of full names (if more than one matches the letters you have typed so far), click on the correct one and it fills in the correct username. As I did with your name above – Will Jagy Dec 26 '20 at 02:51
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    @AndreaMarino Sorry for coming back to this so late! I wasn't meaning to be quite so dismissive with my comment — more just suggesting that while there is a lot of heuristic evidence and conjecture on the density of prime values of polynomials, so little is known for sure that we can't even assert infinitely many primes in the simplest cases. In other words, that your speculation is right and that the question is very hard indeed. – Steven Stadnicki Dec 26 '20 at 21:01
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    @StevenStadnicki: no problem at all :) to be honest, I am very happy with what I have discovered and I thank you all. I'd accept an answer explaining me some of the main heuristic ideas behind BH conjecture, or if this is too much, some of the tools. – Andrea Marino Dec 26 '20 at 22:10

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