3 problems

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

Let $(a_1,a_2,\ldots,a_k)$ be a monotonically increasing sequence of positive even integers. Let $p$ be prime, and let $P=(p,p+a_1,p+a_2,\ldots,p+a_k)$ be an admissible prime $k$-t…

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…