Primes of the form $an^2+bn+c$?

Question

Wondering if this has been proven or disproven. Given:

$a,b,c$ integers

$a$, $b$, and $c$ coprime

$a+b$ and $c$ not both even

$b^2$-$4ac$ not a perfect square

are there infinite primes of the form $an^2 + bn + c$?

Answer 1

Take $a = 1$, $b = 0$, $c = 1$. It is currently open whether there are infinitely many primes of the for $x^2 + 1$.

Answer 2

At the same time, if $\gcd(a,b,c) = 1$ and $b^2 - 4 a c$ is not a square, then the two-variable $$ f(x,y) = a x^2 + b x y + c y^2 $$ does represent an infinite number of primes with $x,y$ integers. This is by Chebotarev Density, see Primes of the Form $x^2 + n y^2$ by David A. Cox, see Theorem 9.12 on page 188, and example on page 190.

If $b^2 - 4ac$ is negative, the form $f(x,y)$ is positive definite (once $a$ is positive). If $b^2 - 4 a c$ is positive, but not a square, the form is indefinite, and in many cases the represented (positive!) primes $p$ do not agree with the primes $q$ for which $(-q)$ is represented. So, for example, $x^2 - 3 y^2$ represents $p$ when $p \equiv 1 \pmod {12},$ but represents $-q$ when $q \equiv 11 \pmod {12}.$

To emphasize, for indefinite forms (not square discriminant) there are both infinitely many primes $p$ represented and infinitely many $q$ for which $-q$ is represented. If you are not worried about the density, there were earlier and easier proofs; see https://mathoverflow.net/questions/144544/primes-represented-by-an-indefinite-binary-quadratic-form

Hmmm. Reading Franz's answer again, it seems Chebotarev may have been first, although existing techniques could have done the full job earlier.