Is the Green-Tao theorem true for primes within a given arithmetic progression? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T06:21:33Z http://mathoverflow.net/feeds/question/25402 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/25402/is-the-green-tao-theorem-true-for-primes-within-a-given-arithmetic-progression Is the Green-Tao theorem true for primes within a given arithmetic progression? Akela 2010-05-20T18:36:26Z 2010-09-14T02:58:41Z <p>Ben Green and Terrence Tao proved that there are arbitrary length arithmetic progressions among the primes.</p> <p>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.</p> <p>Question: Is something known about this stronger statement?</p> http://mathoverflow.net/questions/25402/is-the-green-tao-theorem-true-for-primes-within-a-given-arithmetic-progression/25403#25403 Answer by David Hansen for Is the Green-Tao theorem true for primes within a given arithmetic progression? David Hansen 2010-05-20T18:44:01Z 2010-05-20T18:44:01Z <p>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 $1/\phi(d)$.</p>