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.