Question. Has the following "slackness principle" been tried out before by number theorists? Why not?
Let $X$ be the set of $2k$-prime pair averages for fixed $k\geq 1$. I.e. $X = \{ x \in \Bbb{N} : x \pm 2k \in \Bbb{P}\}$. When $k = 1$, $X$ is the set of twin prime pair averages.
Equivalently...
Question. Has the following "slackness principle" been tried out before by number theorists? Why not?
Let $X$ be the set of $2k$-prime pair averages for fixed $k\geq 1$. I.e. $X = \{ x \in \Bbb{N} : x \pm 2k \in \Bbb{P}\}$. When $k = 1$, $X$ is the set of twin prime pair averages.
Equivalently...
