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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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    This is unknown , but the bunyakovsky-conjecture says "yes". – Peter Mar 20 '18 at 13:13
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    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}. $ – DanielWainfleet Mar 20 '18 at 14:39
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    @DanielWainfleet This conjecture (Catalan's conjecture) has been proven in the meantime. But for other differences , the problem is apparently still open. – Peter Mar 20 '18 at 14:49
  • @Peter . I had not known it was solved. – DanielWainfleet Mar 20 '18 at 15:05
  • Related: https://math.stackexchange.com/questions/2003246/primes-of-the-form-2n2-2n-1/2003612#2003612 – Jack D'Aurizio Mar 20 '18 at 15:12
  • Related post. – Tito Piezas III Mar 20 '18 at 15:14
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    Yes, some results are known https://en.wikipedia.org/wiki/Green%E2%80%93Tao_theorem#Extensions_and_generalizations, but this is not enough for this case ... – rtybase Mar 20 '18 at 21:03

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