Is there a list of all known Sophie Germain prime numbers available anywhere for download? I found a small list from OEIS and the top 20 biggest of such primes, but I can't find a list that would contain numbers that are not on the extremes of size.

  • $\begingroup$ Sorry maybe this is a strange comment, but for what do you need such a list? I just can't imagine that such a list would be useful for anything. $\endgroup$ – Dominic Michaelis Mar 17 '13 at 11:06
  • $\begingroup$ vaxasoftware.com/doc_eduen/mat/primsophie_en.pdf. It is not the list of all the Sophi Germain Primes. $\endgroup$ – Inceptio Mar 17 '13 at 11:44
  • $\begingroup$ Have you thought about how long that list would be? The number of primes $p$ less than $N$ such that $2p+1$ is prime is (heuristically) about $N/(\log N)^2$. It's probably easier to generate as many of these primes as you want on the fly as it is to import a table someone else has made. $\endgroup$ – Gerry Myerson Mar 17 '13 at 12:35
  • $\begingroup$ @DominicMichaelis I want to use those numbers for Pohlig-Hellman cipher. However, I also have to balance that with performance of various machines the algorithm will be deployed on (desktop and mobile). So I need a list of such numbers in order to be able to find the right bit size sweet spot. $\endgroup$ – ThePiachu Mar 17 '13 at 12:48
  • $\begingroup$ @GerryMyerson I would, but a good list of prime numbers is also hard to come by - math.stackexchange.com/q/319570/20958 . $\endgroup$ – ThePiachu Mar 17 '13 at 12:49

Using Gerrys heuristic show us that such a list won't be helpfull at all. With Mathematica you can calculate the $10^9$ prime number in a time which is so small, that mathematica only gives $0.$

The $10^9$th prime number is $22801763489$, with Gerrys heuristic we will have about $4 \cdot 10^7$ Sophie Germain Prime numbers below this prime. Lets assume every of those numbers will need 1 byte in the list, then this list takes $4 \cdot 10^7 $ bytes, which is about 40 mega bytes. This may not seem to be much but they still wasting your cache, and when your list gets larger it won' t fit in the cache and so will extremly slow your computations (taking the sophie primes numbers below the $10^{12}$ primes would be a list of about $3$ giga bytes).

And the list alone doesn't help you, as your program needs to find it in the list, if you give your list a nice structure (using sublists etc) you may get a complextiy of $\mathcal{O}(\log(L))$ to find a number inside your list where $L$ is the number of numbers in your list.

For a lot of problems this will consume much more time than the factorisation itself.

And as you said yourself it is hard to find such a list, this indicates, that it is not so helpful, cause else it would be easier to find.

  • $\begingroup$ Can try writing a python program for the same? Is it helpful? $\endgroup$ – Inceptio Mar 17 '13 at 13:36

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