I've heard that linear polynomials with proper integer coefficients has infinite many positive integers $n$ such that $f(n)$ is prime, by Dirichlet's theorem.

But is there something done with general quadratics or this specific one $f(n)=n^2 +n+1$?

I tried to find some theorems which can be applied to this $f$ but I failed, and solving this problem directly seems too difficult for me.

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    $\begingroup$ This is unknown , but the bunyakovsky-conjecture says "yes". $\endgroup$ – Peter Mar 20 '18 at 13:13
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    $\begingroup$ Research on the number of primes generated by a quadratic goes back a long time, and very little is known. There are many "innocent-sounding" Q's about $\Bbb N$ that are unanswered. One is whether there exist $x,y,a,b\in \Bbb N$ with $a>1$ and $b>1,$ such that $|x^a-y^b|=1,$ other than $\{x^a,y^b\}=\{3^2,2^3\}=\{9,8\}. $ $\endgroup$ – DanielWainfleet Mar 20 '18 at 14:39
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    $\begingroup$ @DanielWainfleet This conjecture (Catalan's conjecture) has been proven in the meantime. But for other differences , the problem is apparently still open. $\endgroup$ – Peter Mar 20 '18 at 14:49
  • $\begingroup$ @Peter . I had not known it was solved. $\endgroup$ – DanielWainfleet Mar 20 '18 at 15:05
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    $\begingroup$ Yes, some results are known en.wikipedia.org/wiki/…, but this is not enough for this case ... $\endgroup$ – rtybase Mar 20 '18 at 21:03

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