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Old 2020-05-17, 23:56   #1
MooMoo2
 
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Default 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=1-750G. 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*2^211088-1 is prime! (63553 decimal digits) Time : 13.336 sec.
1068669447*2^211089-1 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*2^211088-1 are 7,056,154,990,879,113...360,912,313,516,031, and the digits of 1068669447*2^211089-1 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*2^211088-1 and k*2^211089-1 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=1-1.07G for that n, which is now at p=5T with 519,781 candidates remaining.
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Old 2020-05-18, 07:52   #2
pepi37
 
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Quote:
However, I decided against it since the additional sieving may have eliminated Sophies

Explain this please
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Old 2020-05-18, 14:18   #3
VBCurtis
 
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Quote:
Originally Posted by pepi37 View Post
Explain this please
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.
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Old 2020-05-18, 14:41   #4
ATH
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Gratz on the top 10 Sophie Germain prime!
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Old 2020-05-18, 14:56   #5
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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"
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Old 2020-05-18, 22:08   #6
pepi37
 
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Quote:
Originally Posted by VBCurtis View Post
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.
Quote:

However, I decided against it since the additional sieving may have eliminated Sophies

But this can be totally wrong approach since no one know what sieve depth will remove sophie option? :)
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Old 2020-05-19, 00:08   #7
VBCurtis
 
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I have no idea what you are talking about.
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Old 2020-05-19, 10:58   #8
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Quote:
Originally Posted by pepi37 View Post
But this can be totally wrong approach since no one know what sieve depth will remove sophie option? :)
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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Old 2020-06-04, 09:41   #9
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FYI: I've included this k-value into the Wiki and tested the whole range to n=250k.
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