3 problems

The strong Hardy–Littlewood conjecture. For every $\varepsilon>0$,

Consecutive-prime conjecture. For every fixed $\varepsilon>0$,

Let $C_{2r}$ denote the Hardy–Littlewood constant associated with the prime pair $(p,p+2r)$. An arithmetic subsequence of the index sequence $\{2r\}$ means a subsequence obtained b…