Hardy–Littlewood prime $k$-tuples conjecture

Let $h_1,h_2,\ldots,h_k$ be distinct positive even integers such that the numbers in the sequence $\mathbb{P}_k=(p,p+h_1,\ldots,p+h_k)$ do not form a complete residue class modulo any prime. Define

\pi_k(x)=\\#\\{p\leq x:p,p+h_1,\ldots,p+h_k\in\mathbb{P}\}.

Let $w(p;h_1,\ldots,h_k)$ be the number of distinct residues of $0,h_1,\ldots,h_k$ modulo $p$. First Hardy–Littlewood conjecture.

$$\pi_k(x)\sim C_k\int_{2}^{x}\frac{dt}{\log^{k+1}t},$$

where

$$C_k=2^k\prod_{\substack{p\in\mathbb{P}\\\p\\;\mathrm{odd}}}\frac{1-\frac{w(p;h_1,\ldots,h_k)}{q}}{\left(1-\frac{1}{p}\right)^{k+1}}.$$

This is a fundamental conjecture about prime patterns and remains open in general.