The Dickson–Hardy–Littlewood prime-tuples conjecture

A finite set $H\subset\mathbb{N}$ is admissible if, for every prime $p$, there is a residue class modulo $p$ containing no element of $H$.

Dickson–Hardy–Littlewood conjecture. If $H\subset\mathbb{N}$ is finite and admissible, then there exist infinitely many $n\in\mathbb{N}$ such that

$$n+H\subset\mathcal P,$$

where $\mathcal P$ is the set of primes. The paper uses this standard prime-tuples conjecture to derive its bounded finite-sums statement in the primes conditionally. It remains unproved in general.