can Sophie Germain prime be arbitrarily many?

Question

We know that there exists arbitrarily long prime arithmetic progressions by BEN-TAO. Together with Dirichlet's theorem on arithmetic progressions， can we address that Sophie Germain prime number be arbitrarily many? Note that the arithmetic progression of $2*p+1$ ($p$ from arbitrarily long prime arithmetic progressions) follows with Dirichlet's theorem.

Answer 1

No, the Green-Tao theorem is not enough (not even with Dirichlet's theorem) to prove that there are infinitely many Sophie Germain primes. The Sophie Germain pairs have infinite complexity in the Gowers norm and thus current methods do not yet apply to them. For more information see Linear Equations in Primes.

Recent advances (Zhang, Maynard, Polymath) toward the twin prime conjecture makes progress on the Sophie Germain conjecture plausible, but we're not there yet.