Bouniakowsky's conjecture on prime values of irreducible polynomials

Let $f(n)$ be a polynomial with integer coefficients. Assume that its leading coefficient is positive, that $f$ is irreducible over the integers, and that

$$\gcd(f(1),f(2),f(3),\ldots)=1.$$

Bouniakowsky's conjecture. Then $f(n)$ is prime for infinitely many positive integer values of $n$. This is the motivating conjecture for studying eventually prime-free sequences, since reducible or fixed-divisor obstructions can prevent a polynomial sequence from taking prime values infinitely often.