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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.

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This would imply the twin prime conjecture (take $k=1$ and $m$ arbitrarily large). Even if it were only true for $k\ge 1000$, it would imply $\liminf_{n\to\infty}(p_n-p_{n-1})<\infty$, which is a known hard problem. – Anthony Quas Dec 18 '12 at 19:21
up vote 6 down vote accepted

János Pintz considered such questions recently, see his preprints here and here. 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 $d>0$.

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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.

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