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Suppose $p$ is a prime such that $p + 2$ is also prime, and nothing else is known about $p$.

Is there any reason to think that this affects the probability that $p$ is also a Sophie Germain prime? Other than the fact that $p \pmod 6$ would have to be $5$? It would seem that the answer has to be no, but I have some perplexing computational results that I would like to account for.

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    $\begingroup$ What exactly is your numeric evidence? $\endgroup$
    – Alex Degtyarev
    Commented Mar 31, 2014 at 7:01
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    $\begingroup$ It's not just mod 6; there is a bias modulo every small prime. For instance, mod 5, a prime p is equal to 1,2,3,4 mod 5, and needs to avoid 2 mod 5 to have any chance to be a Sophie Germain prime (unless it actually is 2). But if p is a twin prime, it avoids 3 mod 5 (if it isn't actually 3), which slightly raises the chance that it is equal to 2 mod 5 instead. $\endgroup$
    – Terry Tao
    Commented Mar 31, 2014 at 20:01
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    $\begingroup$ The asymptotics from the Hardy-Littlewood prime tuple conjecture can be used to predict the number of tuples of the form p, p+2, 2p+1 that are simultaneously prime, and this can lead to a prediction of the precise bias in the probability that 2p+1 is prime if one conditions on the event that p+2 is prime also. $\endgroup$
    – Terry Tao
    Commented Mar 31, 2014 at 20:03
  • $\begingroup$ See COMMENTS by John W. Nicholson, on Dec 14 2013 at oeis.org/A168421 . $\endgroup$
    – John Nicholson
    Commented Apr 10, 2014 at 2:45

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