Polynomial generalization of propositions regarding the existence of infinitely many primes of a certain form

Question

Many propositions in number theory concerns about the existence of primes of a certain form. Some of them has been proved to be true, such as for $x$ (Euclid's theorem), $4x+1$, etc.; many others remain conjectures, such as for $x^{2} + 1$ (one of the Landau's problems).

A common characteristic of the examples presented above is that the "forms" are all polynomials. One can then ask the more general question: what is the set of "polynomials forms" for which there are infinitely many primes of? In other words, given a random polynomial, how can we tell if infinitely many prime numbers of that form exist?

I am quite sure that this thought has been investigated before since this question is quite natural to ask, but I was not able to find any descriptions of this problem online. Could someone provide some existing approaches, or resources in general?

P.S. I must say that I am an only an amateur on this topic, and do not know about the proper terms professionals use to describe the above-presented vague/awkwardly-stated ideas such as "any polynomial form" or "infinitely many". I would appreciate anyone who would provide a better statement of my thoughts presented above, with which the internet researches could become much more effective.

Many questions posted here on Math Stackexchange are concerned with propositions of the described kind (if infinitely many primes of a "polynomial form" exist). For my question, I think it would be helpful to gather them and study in similarity/differences of their respective proofs. I will list some of them below. — Vincent Phillips, Feb 12 '23 at 19:36
FYI, see Schinzel's hypothesis H, obtained from this answer to Are there infinitely many primes of the form X? We probably don't know.. If you find anything potentially useful, or at least interesting, that's not mentioned in any of the answers to that question, I suggest you consider adding to an appropriate answer there this new information. — John Omielan, Feb 12 '23 at 19:38
First of all take a look at https://en.wikipedia.org/wiki/Bunyakovsky_conjecture — reuns, Feb 12 '23 at 19:44
@JohnOmielan Thanks for the links. I just took a brief look at them and found Are there infinitely many primes of the form X? We probably don't know. quite a satisfactory statement of my question. I might delete this question in the future or edit it so it asks for something similar/more general but also more valuable. — Vincent Phillips, Feb 12 '23 at 19:50

score 1 · Answer 1 · answered Feb 13 '23 at 05:01

Look up the Bateman-Horn conjecture. It explains quantitatively, not just qualitatively, when we expect any finite set of polynomials with integral coefficients has infinitely many simultaneous prime values.

This can be extended to include the expected frequency of prime values of polynomials that are integral-valued on $\mathbf Z$ even if the coefficients are not all integers, such as prime values of $(x^2-x-2)/2$.

Polynomial generalization of propositions regarding the existence of infinitely many primes of a certain form

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