Is the Green-Tao theorem true for primes within a given arithmetic progression?

Question

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?

score 139 · Accepted Answer · answered May 20 '10 at 18:44

139

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 $1/\phi(d)$.

answered May 20 '10 at 18:44

David Hansen

13,018

362

can't argue with that.... – Ben Green May 20 '10 at 19:45
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Neither can I... – Terry Tao May 24 '10 at 06:35
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This is why I love Mathoverflow... – Koundinya Vajjha Sep 19 '10 at 13:34
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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. – plm Dec 04 '12 at 01:24
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@KoundinyaVajjha, I think that another reason to love MO is the exactly 256 upvotes (each) that Ben Green and Terry Tao got. – LSpice Nov 23 '16 at 17:15
@LSpice It continues to this day – FShrike Nov 11 '22 at 15:39

Is the Green-Tao theorem true for primes within a given arithmetic progression?

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