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?



Is there a Cramer's conjecture for Sophie Germain primes?
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.


