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Is $d^2+d+1$ prime for infinitely many $d\in \mathbb{Z}_{>0}$?

This is expected by the Bunyakovsky conjecture which says that, under some conditions, given a polynomial $p(x) \in \mathbb{Z}[x]$ we have $p(d)$ prime for infinitely many $d\in \mathbb{Z}_{>0}$. Is there some proof of this when $p(x) = x^2+x+1$?

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    $\begingroup$ I think there is no polynomial of degree $\ge 2$ for which this is known to hold. $\endgroup$
    – YCor
    Commented Jan 18, 2023 at 13:33
  • $\begingroup$ As mentioned in the section "Partial results: only Dirichlet's theorem" of the Wikipedia page you linked, this is only known for linear polynomials, and for no others. $\endgroup$
    – Boaz Moerman
    Commented Jan 18, 2023 at 13:40
  • $\begingroup$ At least you can prove that if $X$ is an integer greater that $1$ such that $p(X)$ is a prime number then $X \equiv 0 \pmod{3}$ or $X \equiv 1 \pmod{3}$: in other words, the polynomial does produce infinitely many composite numbers. $\endgroup$
    – José Hdz. Stgo.
    Commented Jan 18, 2023 at 13:40
  • $\begingroup$ @JoséHdz.Stgo. This is true for every polynomial though, so it is not a special property of this one. This is for example shown in the answer to math.stackexchange.com/questions/86018/…. $\endgroup$
    – Boaz Moerman
    Commented Jan 18, 2023 at 13:45
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    $\begingroup$ In the given context it's good to recall en.wikipedia.org/wiki/Friedlander%E2%80%93Iwaniec_theorem. $\endgroup$
    – Wlod AA
    Commented Jan 18, 2023 at 14:13

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To answer the question and summarizing the comments: the answer is no, there is no known proof of this conjecture.

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  • $\begingroup$ Thanks for the information. This is surprising to me :) $\endgroup$
    – numberwat
    Commented Jan 19, 2023 at 11:03
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    $\begingroup$ For the closely related question of whether there are infinitely many primes of the form $d^2+1$, see the Wikipedia article on Landau's problems for some references. $\endgroup$
    – Timothy Chow
    Commented Jan 19, 2023 at 13:51
  • $\begingroup$ @number yes as a non-number theorist it is indeed surprising that such a basic statement is not known. However number theorists know that we know embarrassingly little about many things, so wouldn’t be surprised that we don’t know this. $\endgroup$
    – Stanley Yao Xiao
    Commented Jan 19, 2023 at 18:43
  • $\begingroup$ I think DHR sieve can help to prove that the number of the positive integers $n\le$ such that $n^2+n+1$ is prime is at least $c\cdot\dfrac{n}{log(n)log(log(n))}$ for some positive constant $c$ and therefore that sum of their reciprocal diverges at least as fast as $log(log(log(n)))$, and that the similar results hold for any other irreducible quadratic polynomial. $\endgroup$
    – user178594
    Commented Jan 20, 2023 at 15:40

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