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.
[Math] list of all known Sophie Germain prime numbers
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Best Answer
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.