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.



About a possible generalization of Green-Tao's theorem
János Pintz considered such questions recently, see his preprints [here](https://arxiv.org/abs/1002.2899) and [here](https://arxiv.org/abs/1004.1067). In particular, under a weak form of the Elliot-Halberstam conjecture there is an integer $d>0$ such that there are arbitrary long arithmetic progressions of primes $p$ such that $p+d$ is the next prime. Assuming the full conjecture one can take $d\leq 16$, while under a natural strengthening of it one can take any even number $d>0$.


