Given $m,n\in\Bbb N$, assume $2<p_1<\dots<p_n<m$ with each $p_i$ a prime. If $p_i$ are picked randomly, what is average and worst case $d\in2\Bbb N$ such that each of $$p_i+d$$ is a prime and what is average and worst case $d$ if we additionally require if $r<p_1+d$ where $r\in\Bbb N$ with $m\leq r$?
At every given $n\in\Bbb N$, what is minimum $d$ that could be needed? For $n=1$, this is twin prime conjecture.