The Hardy–Littlewood and Bateman–Horn prime-tuples conjecture

Let $f_1(x),\ldots,f_r(x)\in\mathbb{Z}[x]$ and let $f(x)=f_1(x)\cdots f_r(x)$. Assume that the $f_i$ are distinct and irreducible in $\mathbb{Z}[x]$, have positive leading coefficients, and that no prime $\ell$ divides $f(n)$ for every $n\in\mathbb{Z}_{>0}$. Define

$$\pi(f_1,\ldots,f_r;x)=\#\{n\le x\mid f_1(n),\ldots,f_r(n)\text{ are all prime}\}$$

and

$$C(f_1,\ldots,f_r)=\prod_p\left(1-\frac{\nu_f(p)}{p}\right)\left(1-\frac{1}{p}\right)^{-r},$$

where $\nu_f(p)$ is the number of solutions $n\in\mathbb{Z}_p$ of $f(n)\equiv0\pmod p$. The Hardy–Littlewood and Bateman–Horn conjecture. One has

$$\pi(f_1,\ldots,f_r;x)\sim\frac{1}{(\deg f_1)\cdots(\deg f_r)}C(f_1,\ldots,f_r)\frac{x}{(\log x)^r}.$$

This conjecture predicts the asymptotic frequency with which several integral polynomials simultaneously take prime values, and is the general framework invoked later for the distribution of exceptional values. It is open in the stated generality.