One way to isolate the few cases where we did find either a Sophie Germain pair or a twin pair is to look for only the ones with an official announcement: link.

Thanks that helped a lot! So as I understand there was only one Sophie Germain of exponent size 1 290 000?

Does it mean that both 2618163402417*2^1290000-1 and 2618163402417*2^1290001-1 are prime? So I have to do +1 to 1 290 000 in this case to obtain second prime?

Also it was 2016 year when it was found. So for 5 years 2016-2021 there was no other Sophie Germain prime found for exponent 1 290 000? Right?

Do you know (is there a way to find out) how many exponents have been searched so far? And at this exponent size approximately how many exponents (or work units or years) should be searched through to find next Sophie Germain? For example to reach a chance 95% of finding next Sophie Germain how many exponents or work units or years should be searched through?

Also is the K-range searched incrementally? So 2618163402417 is the smallest K such that K * 2^1290000-1 is Sophie Germain? One can be sure that this is the smallest K?

2)
Message boards :
Sophie Germain Prime Search :
Sophie Germain Prime Search
(Message 151325)
Posted 342 days ago by Moytra
Can somebody please explain - if I look at list of found Sophie Germain Primes here and in column "Prime" I see for example number
p = 6438607821915*2^1290000-1
then what it means? It means that p ( 6438607821915*2^1290000-1 ) is prime and 2*p + 1 ( 6438607821915*2^1290001-1 ) (here exponent +1) is also prime?

I'm just confused if I have to subtract one (-1) or to add one (+1) to exponent 1290000 to get second prime.

Because according to project description all three of 2^(n-1)-1, 2^n-1, 2^(n+1)-1 were checked, so I wonder if in that list above I see exponent n = 1290000, then do I have to do -1 or +1 to get second prime? Because both -1 and +1 were checked and any (or both) of those two can be prime.