Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.

Now, consider an arithmetic progression with starting term $a$ and common difference $d$. According to Dirichlet's theorem(suitably strengthened), the primes are "equally distributed" in each residue class modulo $d$. Therefore we imagine that the Green-Tao theorem should still be true if instead of primes we consider only those positive primes that are congruent to $a$ modulo $d$. That is, Green-Tao theorem is true for primes within a given arithmetic progression.

Question: Is something known about this stronger statement?


1 Answer 1


The Green-Tao is true for any subset of the primes of positive relative density; the primes in a fixed arithmetic progression to modulus $d$ have relative density $1/\phi(d)$.

  • 348
    $\begingroup$ can't argue with that.... $\endgroup$
    – Ben Green
    May 20, 2010 at 19:45
  • 349
    $\begingroup$ Neither can I... $\endgroup$
    – Terry Tao
    May 24, 2010 at 6:35
  • 77
    $\begingroup$ This is why I love Mathoverflow... $\endgroup$ Sep 19, 2010 at 13:34
  • 8
    $\begingroup$ Well downvote me if you will, but I think the answer should say that the residue mod $d$ must be coprime to $d$, otherwise there is at most one prime in that residue class, so no nontrivial arithmetic progression in primes in that residue class. $\endgroup$
    – plm
    Dec 4, 2012 at 1:24
  • 7
    $\begingroup$ @KoundinyaVajjha, I think that another reason to love MO is the exactly 256 upvotes (each) that Ben Green and Terry Tao got. $\endgroup$
    – LSpice
    Nov 23, 2016 at 17:15

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