TITLE: About a possible generalization of Green-Tao's theorem
QUESTION [5 upvotes]: Hello,
Let's say that an integer $n$ is $k$-primal if $k$ is its smallest primality radius (i.e non negative integer $r$ such that both $n-r$ and $n+r$ are primes).
I think that for every positive integer $m$ and every non negative integer $k$, there exists an arithmetic progression made of $m$ $k$-primal integers.
Has such a generalization of Green-Tao's theorem been considered so far?
If not, is there a heuristics that would make it quite likely?
Thanks in advance.

REPLY [5 votes]: I am sure many people already know this but just I wanted to mention that the Green-Tao theorem is special case of a more general conjecture of Erdos who asserted that if $(a_n)_{n=1}^\infty \subset \mathbb{N}$ such that $\sum_n \frac {1}{a_n} = \infty$, then the sequence $(a_n)$ contains arbitrarily long arithmetic progressions.