A prime $q$ such that $2q+1$ is also a prime is a Sophie Germain prime.
Cramer's conjecture tells gap between consecutive primes is bound by $O(\log^2p)$.
Is there a similar conjecture for Sophie Germain primes?
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$\begingroup$ @FedorPetrov Could you make it as an answer? $\endgroup$– TurboMay 29, 2022 at 13:41
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Heuristics says that $n$ is Sophie Germain prime with probability roughly $1/\log^2n$. Thus the probability that $C\log^an$×consecutive numbers starting from $n$×are not Sophie Germain primes is about $(1−1/\log^2n)^{C\log^an} \sim e^{−C\log^{a−2} n}$ which is too large when $a<3$ and equals $n^{-C} $ when $a=3$. So, it is natural to predict $O(\log^3p)$ bound.
answered May 29, 2022 at 14:28
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$\begingroup$ Is there an expected prime number theorem for these Sophie Germain primes? Is it just $\frac{n}{\log^2n}$? $\endgroup$– TurboMay 30, 2022 at 17:31
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$\begingroup$ @Turbo with some constant, I guess it should be $2 \prod_p (1-2/p)/(1-1/p)^2$, the product is over odd primes. Maybe it is too naive, though. $\endgroup$ May 31, 2022 at 4:40
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$\begingroup$ is there a closed form or anything close to it? $\endgroup$– TurboMay 31, 2022 at 11:27