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1) Message boards : General discussion : Number of prime numbers found (Message 151547)
Posted 16 hours ago by Ravi Fernando
Call it around 10^33 Zettabytes (ZB) = 10^42 TB. The current estimated total world data storage installation is just under 7 ZB.

Wait, 10^54 bytes to store about 10^26 28-digit numbers? They shouldn't take 10^28 bits each, unless you're storing them in tally marks.
2) Message boards : General discussion : Number of prime numbers found (Message 151545)
Posted 1 day ago by Ravi Fernando
Is there an upper limit N for which we know for all numbers <n whether they are prime or not? If so, where is this upper limit and how quickly is it moving upwards?

Depends on what you mean by "know". Does a prime count as "known" if a computer once checked that it's prime, but this information wasn't stored anywhere? If so, then N is probably at least 2^64, and maybe even an order of magnitude or two larger. But I think most of this information isn't stored anywhere, because it's so easy to recompute on the fly. The difficulty of finding all the primes up to (say) 10^25 isn't that it's hard to check whether a 25-digit number is prime or not; it's that doing anything 10^25 times takes a lot of work. And why bother? Probably no one* will ever care whether 7109827749198234719845111 is prime, and if they do then they can easily check it themselves.

In a different direction, it is known exactly how many primes there are up to 10^28--but this was calculated without listing the primes themselves. See here.

* (except for people reading this)
3) Message boards : 321 Prime Search : 3*2^17748034-1 is prime (Message 151493)
Posted 5 days ago by Ravi Fernando
Great find, McDaWisel! This is the largest (and only top-20) prime found this year so far, on PG or otherwise.
4) Message boards : Wieferich and Wall-Sun-Sun Prime Search : WSS with A = -1 found! (Message 151429)
Posted 10 days ago by Ravi Fernando
Thanks, edited. If only all this information could be found in one place!
5) Message boards : Wieferich and Wall-Sun-Sun Prime Search : WSS with A = -1 found! (Message 151412)
Posted 11 days ago by Ravi Fernando
Amazing find. It appears this is only the tenth twelfth known near-Wall-Sun-Sun prime with |A| = 1:

2 (+/-1)
3 (+1)
5 (+1)
17 (-1)
251 (-1)
733 (+1)
1063 (-1)
123863 (-1)
1677209 (+1)
6336823451747417 (-1)
104868559750360787 (+1)
7665762181374748069 (-1)

List from here. Signs from here--double-checked with WolframAlpha because that file has some known errors, including omitting 17. (Edit: tenth and eleventh from here; thanks JeppeSN.)

Also relevant (and now in need of an update) is JeppeSN's OEIS sequence A339855.
6) Message boards : Sophie Germain Prime Search : Sophie Germain Prime Search (Message 151372)
Posted 13 days ago by Ravi Fernando
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?

All correct. We've searched n=1290000 since 2012 and found one Sophie Germain pair and one twin pair. Prior to that we searched n=666666 from 2009 to 2012 and also found one Sophie Germain pair and one twin pair. (Some n=666666 primes show up with slightly larger exponents due to algebraic simplifications; e.g. 4000004 * 2^666666 - 1 = 1000001 * 2^666668 - 1.) In addition to JeppeSN's links, some of this information is in the subproject status for SGS.

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?

The current Sophie Germain Search has only searched n=666666 and n=1290000, although there was an earlier subproject called the Twin Prime Search (2007-2009, it appears) which searched n=195000 and then n=333333.

Before each range begins, there's a sieving phase where we isolate the k's most likely to produce what we're looking for; for the current search this meant finding the k's that have chances to produce both a SG pair and a twin pair. (In fact it's a bit more complicated than that--it was previously stated that we did a quad sieve, but JeppeSN realized some time ago that it was actually only a triple sieve.) Because of this, we can't guarantee what you ask about the smallest k; there are likely several smaller ones which were omitted because e.g. k*2^1290000 + 1 has a small factor. This has no bearing on whether k*2^1290000 - 1 is a Sophie Germain prime, but if it's obviously not a twin prime, then that makes it less worth the effort to test it.

I don't have the information on how far we'd have to search to likely find a second SG pair, but for what it's worth, we are only planning to search n=1290000 up to k=10^13. We're about 64.5% of the way there after 9 years, so there are still some years left, and it's entirely possibly that we won't find another in this range.
7) Message boards : Number crunching : Sub-project "life" expectancy (Message 151331)
Posted 17 days ago by Ravi Fernando
How long approximately until PPS catches up with PPS-Mega? Is there still time to grab a prime from that project?

My best guess would be 1-1.5 years, depending on TdP and challenges.
8) Message boards : Sophie Germain Prime Search : Sophie Germain Prime Search (Message 151326)
Posted 18 days ago by Ravi Fernando
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.

The list includes all primes found in the SGS project, regardless of whether any of k*2^(n-1)-1, k*2^n+1, k*2^(n+1)-1 are prime. Almost always, none of the three are prime. (The SGS subproject finds lots of primes, but very few of them actually turn out to be Sophie Germain or twin primes.) 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. There you can see that everything comes in pairs with the same value of k.
9) Message boards : Problems and Help : I have found a new prime but it does not appear in my list of prime numbers (Message 151277)
Posted 23 days ago by Ravi Fernando
It just took a bit of time.
I can see it there now.
Congrats !

Congrats yourself!
10) Message boards : Number crunching : Credit Milestones (Message 151171)
Posted 31 days ago by Ravi Fernando

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