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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.

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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.

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  • $\begingroup$ thanks for the answer! It definitely can not prove that there are "infinitely" many Sophie Germain primes. Can it be used for prove that there are "arbitrarily' many such primes? $\endgroup$ – Ocean Yu Oct 22 '14 at 2:25
  • $\begingroup$ @OceanYu: No, because those are the same (by induction). $\endgroup$ – Charles Oct 22 '14 at 2:26
  • $\begingroup$ @Charles hi, just in case and regarding this topic, by chance today I found this paper from 2013, "Relations between the Gauss-Eisenstein prime numbers and their correlation with Sophie Germain primes". The researchers claim that there is a proof of the infinitely many SG primes, through a relation with Gauss-Eisenstein primes, but reading it, it seems to be under the same conditions as you explain in your answer: empslocal.ex.ac.uk/people/staff/mrwatkin/zeta/nardelli2013e.pdf $\endgroup$ – iadvd Aug 28 '15 at 7:59
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    $\begingroup$ @iadvd: The paper has an error on page 6, and generally seems to be at the wrong level for this sort of discovery. Flipping through the paper I see that the proof on pp. 32-33 is also flawed, so I imagine its main result is likewise not correct. If you're interested it might be a good exercise to work through their proof and see if you can discover mistakes. $\endgroup$ – Charles Aug 28 '15 at 12:24
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    $\begingroup$ @iadvd: Well, go through it and skip what you don't understand. You might still catch a logic error. $\endgroup$ – Charles Aug 28 '15 at 12:32

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