Buniakowski's conjecture for quadratic polynomials

Let $f(x)\in\mathbb{Z}[x]$ be a quadratic irreducible polynomial such that the sequence $\\{f(n)\\}_{n\geq 1}$ does not have a common factor. Buniakowski's conjecture. There exist infinitely many primes in the sequence. It is unknown even whether every polynomial satisfying these conditions assumes at least one prime value.