20200517, 23:56  #1 
Aug 2010
2·277 Posts 
Sixth largest known Sophie Germain found after testing less than 16086 candidates!
I was looking for a twin that would be larger than the 59,855 digit twin that I found back in 2015: https://www.mersenneforum.org/showthread.php?t=20340 . I also thought that it would be cool to complete something resembling a "Triple Crown" of primes  one top 500 prime in the list of largest known primes of any form, one top 10 twin, and one top 10 Sophie.
So on April 29, I ran a combined sieve for k*2^n+1 and k*2^(n+1)1, with n=211088 and k=1750G. At p=100T, the odds that a candidate would be twin were 1 in 6.5 million, while the odds that a candidate would be either a twin or a Sophie were 1 in 3.25 million. There were 11,325,824 candidates remaining at that sieve depth, which would have yielded an 82.5% chance of finding a twin, an 82.5% chance of finding a Sophie, and a 97% chance of finding a significant prime pair. My original plan was to sieve it further to p=1P for twins only. However, I decided against it since the additional sieving may have eliminated Sophies and because I was impatient and wanted to get to the fun part. To my great surprise, the following popped out on the 16,085th test: https://primes.utm.edu/primes/page.php?id=130903 1068669447*2^2110881 is prime! (63553 decimal digits) Time : 13.336 sec. 1068669447*2^2110891 is prime! (63554 decimal digits) Time : 13.417 sec. 1068669447*2^211088+1 is not prime. Proth RES64: 633BAB8A8251843D Time : 13.511 sec. My computer actually found the Sophie last week on May 10, less than 3 days after I started the LLR work. But I didn't know about it until this morning, since I never expected to find either a twin or a Sophie that quickly and therefore hadn't bothered to check. I later calculated that the odds of finding any significant prime pair at such a low k on that sieve file were less than 1 in 200. In case anyone's curious, the digits of 1068669447*2^2110881 are 7,056,154,990,879,113...360,912,313,516,031, and the digits of 1068669447*2^2110891 are 14,112,309,981,758,227...721,824,627,032,063. k=1,068,669,447 is likely the smallest k for which k*2^2110881 and k*2^2110891 are prime. But that's not proven, so I'll probably run some tests to determine whether it is or not. I'm sieving k=11.07G for that n, which is now at p=5T with 519,781 candidates remaining. 
20200518, 07:52  #2  
Dec 2011
After milion nines:)
1,249 Posts 
Quote:
Explain this please 

20200518, 14:18  #3 
"Curtis"
Feb 2005
Riverside, CA
2^{2}×3×349 Posts 
If you run a sieve that is set up for both sophie germains and twins, sieving eliminates any candidate where *one* of the types has a factor.
Say, 99999 has a factor for the +1 side; 99999 is eliminated from the sieve file. Well, 99999 and 100000 may both be prime, so a sophie was (possibly) missed. 
20200518, 14:41  #4 
Einyen
Dec 2003
Denmark
B13_{16} Posts 
Gratz on the top 10 Sophie Germain prime!

20200518, 14:56  #5 
Dec 2010
2×3^{2} Posts 
Nice one. I think you had better luck than Daniel Papp when he found that 154798125*2^169690+/1 were twins:
https://primes.utm.edu/bios/page.php?id=373 "A big thanks go to the author of NewPgen and PRP to make it possible me to find a huge twin prime. I used proth only for final primality prooving. You can say it was a big luck and probably you're right. I had only a ~0.6% chance to find such a big twin prime with only 1 computer and 4 months" 
20200518, 22:08  #6  
Dec 2011
After milion nines:)
1249_{10} Posts 
Quote:
Quote:
But this can be totally wrong approach since no one know what sieve depth will remove sophie option? :) 

20200519, 00:08  #7 
"Curtis"
Feb 2005
Riverside, CA
2^{2}×3×349 Posts 
I have no idea what you are talking about.

20200519, 10:58  #8 
"Oliver"
Sep 2017
Porta Westfalica, DE
204_{8} Posts 
No, it's not the depth that removes "Sophie's", its the kind of sieve, i. e. if we only remove the number for which a factor was found, we are always fine. But a sieve for twin primes would remove more. Let's have \(a,b,c \in \mathbb{N}\) with \(a + 2 = b = c  2\) and \(b\) prime, if we now find a factor for both \(a\) and \(c\), we could sieve out \(b\) because it cannot be (part of) a twin prime anymore, although itself is prime.

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