Sixth largest known Sophie Germain found after testing less than 16086 candidates!

MooMoo2 · 2020-05-17, 23:56

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.

pepi37 · 2020-05-18, 07:52

Quote:

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

Explain this please

VBCurtis · 2020-05-18, 14:18

Quote:

Originally Posted by pepi37

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.

ATH · 2020-05-18, 14:41

Gratz on the top 10 Sophie Germain prime!

Bottom Quark · 2020-05-18, 14:56

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"

pepi37 · 2020-05-18, 22:08

Quote:

Originally Posted by VBCurtis

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? :)

VBCurtis · 2020-05-19, 00:08

I have no idea what you are talking about.

kruoli · 2020-05-19, 10:58

Quote:

Originally Posted by pepi37

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.

kar_bon · 2020-06-04, 09:41

FYI: I've included this k-value into the Wiki and tested the whole range to n=250k.

MooMoo2 · 2020-06-07, 08:17

Quote:

Originally Posted by kar_bon

FYI: I've included this k-value into the Wiki and tested the whole range to n=250k.

Nice!

Quote:

Originally Posted by MooMoo2

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.

Testing's complete. k=1,068,669,447 is indeed the smallest k for which k*2^211088-1 and k*2^211089-1 are prime.

Stats:
Range: k=1 - 1.07G
Sieve depth: p=12.59T
Candidates remaining: 488465 tests
n=211088 primes found: 182
n=211089 primes found: 1

Primes found:

Code:

1797009*2^211088-1 is prime! (63551 decimal digits)  Time : 13.839 sec.
1797009*2^211089-1 is not prime.  LLR Res64: 48696F36CDE7E5D7  Time : 14.114 sec.
2428599*2^211088-1 is prime! (63551 decimal digits)  Time : 14.044 sec.
2428599*2^211089-1 is not prime.  LLR Res64: 0E3890C8EA0CAED1  Time : 13.825 sec.
3656079*2^211088-1 is prime! (63551 decimal digits)  Time : 13.807 sec.
3656079*2^211089-1 is not prime.  LLR Res64: 51D66D60F71568CF  Time : 13.609 sec.
4455897*2^211088-1 is prime! (63551 decimal digits)  Time : 13.528 sec.
4455897*2^211089-1 is not prime.  LLR Res64: FB06BA0DFE0AD997  Time : 14.229 sec.
8738937*2^211088-1 is prime! (63551 decimal digits)  Time : 12.476 sec.
8738937*2^211089-1 is not prime.  LLR Res64: F11E65339EB4C79F  Time : 12.343 sec.
18821319*2^211088-1 is prime! (63552 decimal digits)  Time : 13.652 sec.
18821319*2^211089-1 is not prime.  LLR Res64: 77B0E08A6BACCBA8  Time : 14.061 sec.
20987007*2^211088-1 is prime! (63552 decimal digits)  Time : 12.949 sec.
20987007*2^211089-1 is not prime.  LLR Res64: 017D4327C7DFBBC4  Time : 13.034 sec.
22281105*2^211088-1 is prime! (63552 decimal digits)  Time : 12.772 sec.
22281105*2^211089-1 is not prime.  LLR Res64: 57B970897AE4C893  Time : 12.647 sec.
34531749*2^211088-1 is prime! (63552 decimal digits)  Time : 13.911 sec.
34531749*2^211089-1 is not prime.  LLR Res64: CDBD94EA71B822DF  Time : 13.696 sec.
34921725*2^211088-1 is prime! (63552 decimal digits)  Time : 13.355 sec.
34921725*2^211089-1 is not prime.  LLR Res64: 0AA73BBA12238511  Time : 13.731 sec.
51490275*2^211088-1 is prime! (63552 decimal digits)  Time : 13.509 sec.
51490275*2^211089-1 is not prime.  LLR Res64: EDE6ED7518189895  Time : 13.332 sec.
56642715*2^211088-1 is prime! (63552 decimal digits)  Time : 13.543 sec.
56642715*2^211089-1 is not prime.  LLR Res64: EC019CA8362C0E1E  Time : 13.562 sec.
59400879*2^211088-1 is prime! (63552 decimal digits)  Time : 13.951 sec.
59400879*2^211089-1 is not prime.  LLR Res64: 5B3EF03DBE144629  Time : 14.650 sec.
65235825*2^211088-1 is prime! (63552 decimal digits)  Time : 13.735 sec.
65235825*2^211089-1 is not prime.  LLR Res64: F66E2418755E4359  Time : 14.033 sec.
75209265*2^211088-1 is prime! (63552 decimal digits)  Time : 13.232 sec.
75209265*2^211089-1 is not prime.  LLR Res64: ADCF45DF248E1DA5  Time : 13.250 sec.
77249949*2^211088-1 is prime! (63552 decimal digits)  Time : 13.342 sec.
77249949*2^211089-1 is not prime.  LLR Res64: 61C6758A95ABF385  Time : 13.708 sec.
77604987*2^211088-1 is prime! (63552 decimal digits)  Time : 13.765 sec.
77604987*2^211089-1 is not prime.  LLR Res64: D53BE9C7A936AB86  Time : 13.314 sec.
79026057*2^211088-1 is prime! (63552 decimal digits)  Time : 14.253 sec.
79026057*2^211089-1 is not prime.  LLR Res64: 3B2CF1C57C2854EF  Time : 14.330 sec.
82286949*2^211088-1 is prime! (63552 decimal digits)  Time : 14.549 sec.
82286949*2^211089-1 is not prime.  LLR Res64: 58BD69E70EC449A2  Time : 14.862 sec.
86801655*2^211088-1 is prime! (63552 decimal digits)  Time : 14.568 sec.
86801655*2^211089-1 is not prime.  LLR Res64: 65A8FEAFAFC403A3  Time : 13.959 sec.
99930297*2^211088-1 is prime! (63552 decimal digits)  Time : 13.463 sec.
99930297*2^211089-1 is not prime.  LLR Res64: 9A91E6FBCA24382B  Time : 13.928 sec.
102053595*2^211088-1 is prime! (63552 decimal digits)  Time : 15.288 sec.
102053595*2^211089-1 is not prime.  LLR Res64: 17544A2410965CBA  Time : 14.401 sec.
104690859*2^211088-1 is prime! (63552 decimal digits)  Time : 13.654 sec.
104690859*2^211089-1 is not prime.  LLR Res64: C13882FC6B244F18  Time : 13.678 sec.
105661185*2^211088-1 is prime! (63552 decimal digits)  Time : 13.743 sec.
105661185*2^211089-1 is not prime.  LLR Res64: CAFEA31ECFBE322F  Time : 13.479 sec.
115607259*2^211088-1 is prime! (63552 decimal digits)  Time : 13.803 sec.
115607259*2^211089-1 is not prime.  LLR Res64: C4F48D67C087CC9B  Time : 13.716 sec.
115945017*2^211088-1 is prime! (63552 decimal digits)  Time : 13.734 sec.
115945017*2^211089-1 is not prime.  LLR Res64: 8A156C0E392442D9  Time : 14.217 sec.
121885197*2^211088-1 is prime! (63552 decimal digits)  Time : 12.120 sec.
121885197*2^211089-1 is not prime.  LLR Res64: C3D698BEBE89E130  Time : 12.144 sec.
122285835*2^211088-1 is prime! (63552 decimal digits)  Time : 13.693 sec.
122285835*2^211089-1 is not prime.  LLR Res64: 9C787299ACFB3BBA  Time : 13.518 sec.
125094045*2^211088-1 is prime! (63552 decimal digits)  Time : 13.814 sec.
125094045*2^211089-1 is not prime.  LLR Res64: 22DB09573E83DA39  Time : 13.253 sec.
142691457*2^211088-1 is prime! (63552 decimal digits)  Time : 15.126 sec.
142691457*2^211089-1 is not prime.  LLR Res64: 42F323184A29C7D2  Time : 15.251 sec.
151184025*2^211088-1 is prime! (63552 decimal digits)  Time : 13.936 sec.
151184025*2^211089-1 is not prime.  LLR Res64: D25C6CBD7FA8A923  Time : 13.689 sec.
160854945*2^211088-1 is prime! (63553 decimal digits)  Time : 14.250 sec.
160854945*2^211089-1 is not prime.  LLR Res64: F194CDD8145DC3F9  Time : 14.130 sec.
164386605*2^211088-1 is prime! (63553 decimal digits)  Time : 14.774 sec.
164386605*2^211089-1 is not prime.  LLR Res64: CDD49AEB0D8CBA1D  Time : 14.546 sec.
174544905*2^211088-1 is prime! (63553 decimal digits)  Time : 13.719 sec.
174544905*2^211089-1 is not prime.  LLR Res64: 3793E0137B10901E  Time : 13.813 sec.
174896529*2^211088-1 is prime! (63553 decimal digits)  Time : 14.469 sec.
174896529*2^211089-1 is not prime.  LLR Res64: 8C5DD6472C8045DA  Time : 13.980 sec.
181891077*2^211088-1 is prime! (63553 decimal digits)  Time : 14.487 sec.
181891077*2^211089-1 is not prime.  LLR Res64: DC945D3AFFAE054C  Time : 14.219 sec.
189757827*2^211088-1 is prime! (63553 decimal digits)  Time : 13.631 sec.
189757827*2^211089-1 is not prime.  LLR Res64: 0E5B8F2C6C782338  Time : 13.653 sec.
196697169*2^211088-1 is prime! (63553 decimal digits)  Time : 13.850 sec.
196697169*2^211089-1 is not prime.  LLR Res64: 30EE58857C7843AF  Time : 13.737 sec.
198769035*2^211088-1 is prime! (63553 decimal digits)  Time : 13.829 sec.
198769035*2^211089-1 is not prime.  LLR Res64: 8065B1A6D9475008  Time : 14.521 sec.
201784635*2^211088-1 is prime! (63553 decimal digits)  Time : 13.654 sec.
201784635*2^211089-1 is not prime.  LLR Res64: 5B20BF2500069B1F  Time : 14.325 sec.
201951837*2^211088-1 is prime! (63553 decimal digits)  Time : 14.432 sec.
201951837*2^211089-1 is not prime.  LLR Res64: B64AC46C3D9C521B  Time : 13.990 sec.
205895715*2^211088-1 is prime! (63553 decimal digits)  Time : 13.769 sec.
205895715*2^211089-1 is not prime.  LLR Res64: 0B4CCD127EEDE5BF  Time : 14.280 sec.
210293937*2^211088-1 is prime! (63553 decimal digits)  Time : 13.847 sec.
210293937*2^211089-1 is not prime.  LLR Res64: 508DF4F5B02E81BB  Time : 14.075 sec.
211922907*2^211088-1 is prime! (63553 decimal digits)  Time : 14.206 sec.
211922907*2^211089-1 is not prime.  LLR Res64: 8F5A46D6CE977871  Time : 14.001 sec.
215964129*2^211088-1 is prime! (63553 decimal digits)  Time : 14.873 sec.
215964129*2^211089-1 is not prime.  LLR Res64: E0DC12526D093AE3  Time : 15.271 sec.
222149469*2^211088-1 is prime! (63553 decimal digits)  Time : 15.799 sec.
222149469*2^211089-1 is not prime.  LLR Res64: 76A0C4EB51F41173  Time : 15.935 sec.
225009309*2^211088-1 is prime! (63553 decimal digits)  Time : 14.319 sec.
225009309*2^211089-1 is not prime.  LLR Res64: 3A0084285FBB7D69  Time : 13.658 sec.
229584837*2^211088-1 is prime! (63553 decimal digits)  Time : 14.468 sec.
229584837*2^211089-1 is not prime.  LLR Res64: A6C08DC55451F90A  Time : 13.937 sec.
239217165*2^211088-1 is prime! (63553 decimal digits)  Time : 14.576 sec.
239217165*2^211089-1 is not prime.  LLR Res64: 6F2F5CEA5D519F7D  Time : 14.544 sec.
241340385*2^211088-1 is prime! (63553 decimal digits)  Time : 14.286 sec.
241340385*2^211089-1 is not prime.  LLR Res64: C44C116C71F8CC57  Time : 14.141 sec.
241453275*2^211088-1 is prime! (63553 decimal digits)  Time : 14.545 sec.
241453275*2^211089-1 is not prime.  LLR Res64: C4AC1F5772230487  Time : 14.288 sec.
246186135*2^211088-1 is prime! (63553 decimal digits)  Time : 14.169 sec.
246186135*2^211089-1 is not prime.  LLR Res64: 9FEAC4F307EA4B44  Time : 13.970 sec.
249768207*2^211088-1 is prime! (63553 decimal digits)  Time : 13.716 sec.
249768207*2^211089-1 is not prime.  LLR Res64: FA8EC17170D5D562  Time : 13.990 sec.
258107217*2^211088-1 is prime! (63553 decimal digits)  Time : 14.617 sec.
258107217*2^211089-1 is not prime.  LLR Res64: 7881F05FC2630729  Time : 13.773 sec.
258680379*2^211088-1 is prime! (63553 decimal digits)  Time : 14.072 sec.
258680379*2^211089-1 is not prime.  LLR Res64: C7C7A8E6B81C4335  Time : 13.783 sec.
258909075*2^211088-1 is prime! (63553 decimal digits)  Time : 15.962 sec.
258909075*2^211089-1 is not prime.  LLR Res64: 92E7F3CA335F1179  Time : 14.161 sec.
261437007*2^211088-1 is prime! (63553 decimal digits)  Time : 14.102 sec.
261437007*2^211089-1 is not prime.  LLR Res64: 36FDBC58ACC0F458  Time : 14.068 sec.
272679867*2^211088-1 is prime! (63553 decimal digits)  Time : 12.398 sec.
272679867*2^211089-1 is not prime.  LLR Res64: 6A9D89447DA597E5  Time : 12.426 sec.
283183437*2^211088-1 is prime! (63553 decimal digits)  Time : 13.617 sec.
283183437*2^211089-1 is not prime.  LLR Res64: E69B44C8AAD964ED  Time : 13.748 sec.
284888187*2^211088-1 is prime! (63553 decimal digits)  Time : 14.736 sec.
284888187*2^211089-1 is not prime.  LLR Res64: 6968B996E79DD3E9  Time : 14.236 sec.
290094207*2^211088-1 is prime! (63553 decimal digits)  Time : 13.886 sec.
290094207*2^211089-1 is not prime.  LLR Res64: 8BFD610ADDBB992A  Time : 13.816 sec.
293738415*2^211088-1 is prime! (63553 decimal digits)  Time : 14.240 sec.
293738415*2^211089-1 is not prime.  LLR Res64: CE1B0C0395EB2E24  Time : 14.015 sec.
297177969*2^211088-1 is prime! (63553 decimal digits)  Time : 14.706 sec.
297177969*2^211089-1 is not prime.  LLR Res64: CAF12587B6177F16  Time : 14.355 sec.
301097037*2^211088-1 is prime! (63553 decimal digits)  Time : 13.833 sec.
301097037*2^211089-1 is not prime.  LLR Res64: D44003AF13350805  Time : 13.661 sec.
302296137*2^211088-1 is prime! (63553 decimal digits)  Time : 13.480 sec.
302296137*2^211089-1 is not prime.  LLR Res64: 4E23031C92E482F8  Time : 13.515 sec.
317571177*2^211088-1 is prime! (63553 decimal digits)  Time : 14.990 sec.
317571177*2^211089-1 is not prime.  LLR Res64: 598A05BF6043E34F  Time : 14.435 sec.
329781189*2^211088-1 is prime! (63553 decimal digits)  Time : 13.088 sec.
329781189*2^211089-1 is not prime.  LLR Res64: CD4845894FEFF78A  Time : 12.881 sec.
332187687*2^211088-1 is prime! (63553 decimal digits)  Time : 14.177 sec.
332187687*2^211089-1 is not prime.  LLR Res64: 39F23D0E84F1DF53  Time : 14.364 sec.
339285867*2^211088-1 is prime! (63553 decimal digits)  Time : 13.274 sec.
339285867*2^211089-1 is not prime.  LLR Res64: C9F7310F7CFE3F78  Time : 13.554 sec.
341877477*2^211088-1 is prime! (63553 decimal digits)  Time : 12.310 sec.
341877477*2^211089-1 is not prime.  LLR Res64: 0B47D7A9B2B7AF54  Time : 12.250 sec.
344451555*2^211088-1 is prime! (63553 decimal digits)  Time : 14.687 sec.
344451555*2^211089-1 is not prime.  LLR Res64: E18A75DCF7E846CD  Time : 14.198 sec.
348898899*2^211088-1 is prime! (63553 decimal digits)  Time : 15.449 sec.
348898899*2^211089-1 is not prime.  LLR Res64: B926069DA6CC835E  Time : 14.906 sec.
349743357*2^211088-1 is prime! (63553 decimal digits)  Time : 15.488 sec.
349743357*2^211089-1 is not prime.  LLR Res64: 35E4A54C03F09061  Time : 15.310 sec.
359106429*2^211088-1 is prime! (63553 decimal digits)  Time : 13.533 sec.
359106429*2^211089-1 is not prime.  LLR Res64: 6E2EA05177650730  Time : 13.844 sec.
368890575*2^211088-1 is prime! (63553 decimal digits)  Time : 15.112 sec.
368890575*2^211089-1 is not prime.  LLR Res64: A663967836A3AC07  Time : 15.668 sec.
376085715*2^211088-1 is prime! (63553 decimal digits)  Time : 14.118 sec.
376085715*2^211089-1 is not prime.  LLR Res64: B19E7F536E4F937F  Time : 13.973 sec.
379900959*2^211088-1 is prime! (63553 decimal digits)  Time : 13.822 sec.
379900959*2^211089-1 is not prime.  LLR Res64: 9DA73F129F508903  Time : 14.542 sec.
384008019*2^211088-1 is prime! (63553 decimal digits)  Time : 13.652 sec.
384008019*2^211089-1 is not prime.  LLR Res64: E7507CE5E5EF5B23  Time : 13.634 sec.
385837197*2^211088-1 is prime! (63553 decimal digits)  Time : 14.004 sec.
385837197*2^211089-1 is not prime.  LLR Res64: F0D9424AE3E78DD8  Time : 13.876 sec.
391435755*2^211088-1 is prime! (63553 decimal digits)  Time : 14.000 sec.
391435755*2^211089-1 is not prime.  LLR Res64: BA6CAA8E0B6AE049  Time : 13.910 sec.
399989769*2^211088-1 is prime! (63553 decimal digits)  Time : 13.978 sec.
399989769*2^211089-1 is not prime.  LLR Res64: 8930C246ADA3E990  Time : 14.767 sec.
423042309*2^211088-1 is prime! (63553 decimal digits)  Time : 15.097 sec.
423042309*2^211089-1 is not prime.  LLR Res64: D010387C4E2C2054  Time : 15.055 sec.
427746177*2^211088-1 is prime! (63553 decimal digits)  Time : 13.747 sec.
427746177*2^211089-1 is not prime.  LLR Res64: 287BAFD0972B739B  Time : 13.882 sec.
429533745*2^211088-1 is prime! (63553 decimal digits)  Time : 13.522 sec.
429533745*2^211089-1 is not prime.  LLR Res64: CBA91FC25B4902DD  Time : 13.559 sec.
431536389*2^211088-1 is prime! (63553 decimal digits)  Time : 14.195 sec.
431536389*2^211089-1 is not prime.  LLR Res64: AE4002BD47EDC3CF  Time : 14.281 sec.
438129627*2^211088-1 is prime! (63553 decimal digits)  Time : 14.041 sec.
438129627*2^211089-1 is not prime.  LLR Res64: B92C3989B913888E  Time : 13.914 sec.
442791975*2^211088-1 is prime! (63553 decimal digits)  Time : 14.001 sec.
442791975*2^211089-1 is not prime.  LLR Res64: 8E11CFA89C57F636  Time : 14.772 sec.
443069337*2^211088-1 is prime! (63553 decimal digits)  Time : 14.414 sec.
443069337*2^211089-1 is not prime.  LLR Res64: 660D53AFF627B345  Time : 14.926 sec.
457354125*2^211088-1 is prime! (63553 decimal digits)  Time : 13.652 sec.
457354125*2^211089-1 is not prime.  LLR Res64: 57A5DA6EFFCD9FDF  Time : 14.327 sec.
468365985*2^211088-1 is prime! (63553 decimal digits)  Time : 13.943 sec.
468365985*2^211089-1 is not prime.  LLR Res64: FAEA3C2A60D79586  Time : 13.795 sec.
470270709*2^211088-1 is prime! (63553 decimal digits)  Time : 14.387 sec.
470270709*2^211089-1 is not prime.  LLR Res64: C74FC4E8662D6F3C  Time : 14.713 sec.
489602487*2^211088-1 is prime! (63553 decimal digits)  Time : 15.073 sec.
489602487*2^211089-1 is not prime.  LLR Res64: 9C9848E4601A5D59  Time : 15.806 sec.
490722249*2^211088-1 is prime! (63553 decimal digits)  Time : 15.097 sec.
490722249*2^211089-1 is not prime.  LLR Res64: 5B0A9AA452060372  Time : 14.413 sec.
513280149*2^211088-1 is prime! (63553 decimal digits)  Time : 13.984 sec.
513280149*2^211089-1 is not prime.  LLR Res64: BAA2ECF3DAB549A7  Time : 13.449 sec.
518492859*2^211088-1 is prime! (63553 decimal digits)  Time : 13.657 sec.
518492859*2^211089-1 is not prime.  LLR Res64: C2636AE85A8FF087  Time : 13.557 sec.
520488345*2^211088-1 is prime! (63553 decimal digits)  Time : 14.328 sec.
520488345*2^211089-1 is not prime.  LLR Res64: 0A9CB655C7CE4603  Time : 14.186 sec.
521415249*2^211088-1 is prime! (63553 decimal digits)  Time : 14.629 sec.
521415249*2^211089-1 is not prime.  LLR Res64: 417877D23856D7DA  Time : 14.351 sec.
532266837*2^211088-1 is prime! (63553 decimal digits)  Time : 12.854 sec.
532266837*2^211089-1 is not prime.  LLR Res64: 23A5E1CAD2AF837B  Time : 12.678 sec.
540685305*2^211088-1 is prime! (63553 decimal digits)  Time : 12.601 sec.
540685305*2^211089-1 is not prime.  LLR Res64: 229D112C6BE4CAFD  Time : 12.416 sec.
545812809*2^211088-1 is prime! (63553 decimal digits)  Time : 12.244 sec.
545812809*2^211089-1 is not prime.  LLR Res64: B14DDBB1371630D6  Time : 12.422 sec.
583030875*2^211088-1 is prime! (63553 decimal digits)  Time : 14.287 sec.
583030875*2^211089-1 is not prime.  LLR Res64: 4FE117E46580CE0B  Time : 14.292 sec.
595403289*2^211088-1 is prime! (63553 decimal digits)  Time : 13.261 sec.
595403289*2^211089-1 is not prime.  LLR Res64: 80898E05A0EFCF8E  Time : 14.174 sec.
602311755*2^211088-1 is prime! (63553 decimal digits)  Time : 14.372 sec.
602311755*2^211089-1 is not prime.  LLR Res64: 61922C35AFB5BE28  Time : 13.940 sec.
603823065*2^211088-1 is prime! (63553 decimal digits)  Time : 14.926 sec.
603823065*2^211089-1 is not prime.  LLR Res64: A2AACA7857ABF4C4  Time : 15.875 sec.
614395449*2^211088-1 is prime! (63553 decimal digits)  Time : 14.357 sec.
614395449*2^211089-1 is not prime.  LLR Res64: 603433FB87F5C401  Time : 13.920 sec.
627271329*2^211088-1 is prime! (63553 decimal digits)  Time : 13.763 sec.
627271329*2^211089-1 is not prime.  LLR Res64: CA7E688D4660783D  Time : 14.389 sec.
628385835*2^211088-1 is prime! (63553 decimal digits)  Time : 13.400 sec.
628385835*2^211089-1 is not prime.  LLR Res64: 0C7EA0B916D65379  Time : 14.123 sec.
628632297*2^211088-1 is prime! (63553 decimal digits)  Time : 14.133 sec.
628632297*2^211089-1 is not prime.  LLR Res64: 7C49B12BE66066B7  Time : 15.233 sec.
634440939*2^211088-1 is prime! (63553 decimal digits)  Time : 14.715 sec.
634440939*2^211089-1 is not prime.  LLR Res64: E488999EA1FA2FF2  Time : 13.699 sec.
637494249*2^211088-1 is prime! (63553 decimal digits)  Time : 13.986 sec.
637494249*2^211089-1 is not prime.  LLR Res64: 29AEA65810C64BA8  Time : 14.027 sec.
642422469*2^211088-1 is prime! (63553 decimal digits)  Time : 14.701 sec.
642422469*2^211089-1 is not prime.  LLR Res64: 2AF574D220BBFD44  Time : 14.766 sec.
649997025*2^211088-1 is prime! (63553 decimal digits)  Time : 13.807 sec.
649997025*2^211089-1 is not prime.  LLR Res64: 7AF26119DE119258  Time : 14.343 sec.
650049147*2^211088-1 is prime! (63553 decimal digits)  Time : 14.213 sec.
650049147*2^211089-1 is not prime.  LLR Res64: 58E53880E1EC3A5E  Time : 14.240 sec.
650561895*2^211088-1 is prime! (63553 decimal digits)  Time : 14.141 sec.
650561895*2^211089-1 is not prime.  LLR Res64: 136DFE6D13A18201  Time : 13.777 sec.
652970505*2^211088-1 is prime! (63553 decimal digits)  Time : 13.645 sec.
652970505*2^211089-1 is not prime.  LLR Res64: A84DCFF6FCECC45C  Time : 13.714 sec.
659620695*2^211088-1 is prime! (63553 decimal digits)  Time : 13.893 sec.
659620695*2^211089-1 is not prime.  LLR Res64: 206E28191F7EDFC6  Time : 14.145 sec.
671660589*2^211088-1 is prime! (63553 decimal digits)  Time : 13.816 sec.
671660589*2^211089-1 is not prime.  LLR Res64: 748EBA63B9002095  Time : 13.804 sec.
680747925*2^211088-1 is prime! (63553 decimal digits)  Time : 13.164 sec.
680747925*2^211089-1 is not prime.  LLR Res64: E850C2CBB5F68E9B  Time : 13.026 sec.
684694827*2^211088-1 is prime! (63553 decimal digits)  Time : 14.567 sec.
684694827*2^211089-1 is not prime.  LLR Res64: B418B4FA86B8CAD5  Time : 14.645 sec.
690947025*2^211088-1 is prime! (63553 decimal digits)  Time : 14.692 sec.
690947025*2^211089-1 is not prime.  LLR Res64: 263D6A44EDA61865  Time : 14.813 sec.
705309399*2^211088-1 is prime! (63553 decimal digits)  Time : 13.662 sec.
705309399*2^211089-1 is not prime.  LLR Res64: F34D327E73565342  Time : 13.783 sec.
718530999*2^211088-1 is prime! (63553 decimal digits)  Time : 13.528 sec.
718530999*2^211089-1 is not prime.  LLR Res64: A9ECE3A786159254  Time : 13.648 sec.
728475885*2^211088-1 is prime! (63553 decimal digits)  Time : 13.837 sec.
728475885*2^211089-1 is not prime.  LLR Res64: 4671B89E78AD046C  Time : 14.310 sec.
730552785*2^211088-1 is prime! (63553 decimal digits)  Time : 14.046 sec.
730552785*2^211089-1 is not prime.  LLR Res64: 33C8F4DB58C35850  Time : 14.093 sec.
732985029*2^211088-1 is prime! (63553 decimal digits)  Time : 13.564 sec.
732985029*2^211089-1 is not prime.  LLR Res64: 7734D0D13D1345A4  Time : 14.190 sec.
735291939*2^211088-1 is prime! (63553 decimal digits)  Time : 13.322 sec.
735291939*2^211089-1 is not prime.  LLR Res64: 4F317F4FB3A8D7E0  Time : 12.815 sec.
735706515*2^211088-1 is prime! (63553 decimal digits)  Time : 13.251 sec.
735706515*2^211089-1 is not prime.  LLR Res64: 9BB9E27553E1CB99  Time : 13.093 sec.
737875599*2^211088-1 is prime! (63553 decimal digits)  Time : 14.554 sec.
737875599*2^211089-1 is not prime.  LLR Res64: 0B02AA5555B2E988  Time : 14.714 sec.
742167249*2^211088-1 is prime! (63553 decimal digits)  Time : 14.499 sec.
742167249*2^211089-1 is not prime.  LLR Res64: 86D27840490D5E83  Time : 14.131 sec.
758647125*2^211088-1 is prime! (63553 decimal digits)  Time : 14.385 sec.
758647125*2^211089-1 is not prime.  LLR Res64: 3426968FBE64598E  Time : 13.969 sec.
759850917*2^211088-1 is prime! (63553 decimal digits)  Time : 13.880 sec.
759850917*2^211089-1 is not prime.  LLR Res64: 1CE63CC2D6FA90F6  Time : 13.735 sec.
760079715*2^211088-1 is prime! (63553 decimal digits)  Time : 13.647 sec.
760079715*2^211089-1 is not prime.  LLR Res64: 6132E7CC4D6C152F  Time : 13.420 sec.
766213059*2^211088-1 is prime! (63553 decimal digits)  Time : 13.700 sec.
766213059*2^211089-1 is not prime.  LLR Res64: 9626F605B3367FB4  Time : 13.666 sec.
777039099*2^211088-1 is prime! (63553 decimal digits)  Time : 13.605 sec.
777039099*2^211089-1 is not prime.  LLR Res64: 5C25E4498C4D0810  Time : 13.835 sec.
784651515*2^211088-1 is prime! (63553 decimal digits)  Time : 13.609 sec.
784651515*2^211089-1 is not prime.  LLR Res64: 49EB0F3E3C489CA7  Time : 13.398 sec.
789487797*2^211088-1 is prime! (63553 decimal digits)  Time : 13.792 sec.
789487797*2^211089-1 is not prime.  LLR Res64: B3BD39D0EE81AC49  Time : 14.461 sec.
790738539*2^211088-1 is prime! (63553 decimal digits)  Time : 13.768 sec.
790738539*2^211089-1 is not prime.  LLR Res64: EE2D6FCEF97E83FE  Time : 13.707 sec.
793188429*2^211088-1 is prime! (63553 decimal digits)  Time : 14.033 sec.
793188429*2^211089-1 is not prime.  LLR Res64: E4509BED8410C6F2  Time : 14.027 sec.
800674257*2^211088-1 is prime! (63553 decimal digits)  Time : 14.044 sec.
800674257*2^211089-1 is not prime.  LLR Res64: 304F4D8504464DBF  Time : 13.707 sec.
801396459*2^211088-1 is prime! (63553 decimal digits)  Time : 12.491 sec.
801396459*2^211089-1 is not prime.  LLR Res64: 9C08298CC7D9F335  Time : 12.638 sec.
808322259*2^211088-1 is prime! (63553 decimal digits)  Time : 12.377 sec.
808322259*2^211089-1 is not prime.  LLR Res64: 3B3735382EC5125B  Time : 12.512 sec.
813093837*2^211088-1 is prime! (63553 decimal digits)  Time : 12.181 sec.
813093837*2^211089-1 is not prime.  LLR Res64: 0F6C56AEACE235A1  Time : 12.449 sec.
823175757*2^211088-1 is prime! (63553 decimal digits)  Time : 14.041 sec.
823175757*2^211089-1 is not prime.  LLR Res64: DE8DF925E748999C  Time : 13.884 sec.
831145419*2^211088-1 is prime! (63553 decimal digits)  Time : 13.903 sec.
831145419*2^211089-1 is not prime.  LLR Res64: 6C8D744C375DDEB0  Time : 14.119 sec.
837729975*2^211088-1 is prime! (63553 decimal digits)  Time : 13.729 sec.
837729975*2^211089-1 is not prime.  LLR Res64: 51C1A746683DF5C4  Time : 13.932 sec.
841118967*2^211088-1 is prime! (63553 decimal digits)  Time : 13.779 sec.
841118967*2^211089-1 is not prime.  LLR Res64: FE32768C13E942DC  Time : 13.758 sec.
850940835*2^211088-1 is prime! (63553 decimal digits)  Time : 14.105 sec.
850940835*2^211089-1 is not prime.  LLR Res64: 6353DD5A48192A95  Time : 13.706 sec.
864952347*2^211088-1 is prime! (63553 decimal digits)  Time : 13.754 sec.
864952347*2^211089-1 is not prime.  LLR Res64: E6D1D728616C113B  Time : 13.822 sec.
871342167*2^211088-1 is prime! (63553 decimal digits)  Time : 13.437 sec.
871342167*2^211089-1 is not prime.  LLR Res64: 0DAA03F146C2FAD4  Time : 13.209 sec.
873981567*2^211088-1 is prime! (63553 decimal digits)  Time : 14.176 sec.
873981567*2^211089-1 is not prime.  LLR Res64: 999450199C474355  Time : 14.002 sec.
875731737*2^211088-1 is prime! (63553 decimal digits)  Time : 13.977 sec.
875731737*2^211089-1 is not prime.  LLR Res64: A68BAD12219A17FE  Time : 13.973 sec.
877595577*2^211088-1 is prime! (63553 decimal digits)  Time : 12.141 sec.
877595577*2^211089-1 is not prime.  LLR Res64: 6C15D977E94D51A4  Time : 12.348 sec.
880815579*2^211088-1 is prime! (63553 decimal digits)  Time : 13.864 sec.
880815579*2^211089-1 is not prime.  LLR Res64: 9C48E8F6121016E8  Time : 14.410 sec.
881894349*2^211088-1 is prime! (63553 decimal digits)  Time : 14.751 sec.
881894349*2^211089-1 is not prime.  LLR Res64: 9F33E0EA203432E0  Time : 15.058 sec.
884786397*2^211088-1 is prime! (63553 decimal digits)  Time : 13.776 sec.
884786397*2^211089-1 is not prime.  LLR Res64: 306C6D9AB1D48B33  Time : 14.363 sec.
887407449*2^211088-1 is prime! (63553 decimal digits)  Time : 14.286 sec.
887407449*2^211089-1 is not prime.  LLR Res64: FA24C85963B9FB0D  Time : 14.340 sec.
889218447*2^211088-1 is prime! (63553 decimal digits)  Time : 14.283 sec.
889218447*2^211089-1 is not prime.  LLR Res64: 09C29B09457CC344  Time : 13.368 sec.
892457055*2^211088-1 is prime! (63553 decimal digits)  Time : 13.284 sec.
892457055*2^211089-1 is not prime.  LLR Res64: BB2B43C1AB2F0E08  Time : 13.642 sec.
904097445*2^211088-1 is prime! (63553 decimal digits)  Time : 13.103 sec.
904097445*2^211089-1 is not prime.  LLR Res64: 01782D1E3F706A84  Time : 13.368 sec.
916712109*2^211088-1 is prime! (63553 decimal digits)  Time : 14.308 sec.
916712109*2^211089-1 is not prime.  LLR Res64: 8066F78A3C10504E  Time : 13.763 sec.
929893689*2^211088-1 is prime! (63553 decimal digits)  Time : 14.048 sec.
929893689*2^211089-1 is not prime.  LLR Res64: 96BC75B564BB9E40  Time : 13.929 sec.
951550509*2^211088-1 is prime! (63553 decimal digits)  Time : 14.498 sec.
951550509*2^211089-1 is not prime.  LLR Res64: E63E3D65238A1AFA  Time : 13.640 sec.
955322067*2^211088-1 is prime! (63553 decimal digits)  Time : 13.929 sec.
955322067*2^211089-1 is not prime.  LLR Res64: 3F26BE6581A39840  Time : 14.014 sec.
958094319*2^211088-1 is prime! (63553 decimal digits)  Time : 15.561 sec.
958094319*2^211089-1 is not prime.  LLR Res64: 4639F2F1F924CDA8  Time : 15.923 sec.
962429355*2^211088-1 is prime! (63553 decimal digits)  Time : 14.284 sec.
962429355*2^211089-1 is not prime.  LLR Res64: EA352D2E7D81A8B8  Time : 14.014 sec.
975538485*2^211088-1 is prime! (63553 decimal digits)  Time : 14.196 sec.
975538485*2^211089-1 is not prime.  LLR Res64: C84501A6A2189F30  Time : 13.672 sec.
981530757*2^211088-1 is prime! (63553 decimal digits)  Time : 13.495 sec.
981530757*2^211089-1 is not prime.  LLR Res64: 30781B0D52B1AA42  Time : 14.101 sec.
981889887*2^211088-1 is prime! (63553 decimal digits)  Time : 14.108 sec.
981889887*2^211089-1 is not prime.  LLR Res64: 3FBCB3966114A80B  Time : 14.396 sec.
982094529*2^211088-1 is prime! (63553 decimal digits)  Time : 14.552 sec.
982094529*2^211089-1 is not prime.  LLR Res64: 5CA9FCE201CBB438  Time : 14.433 sec.
983268369*2^211088-1 is prime! (63553 decimal digits)  Time : 14.062 sec.
983268369*2^211089-1 is not prime.  LLR Res64: 41CC94883C4FA213  Time : 13.932 sec.
1005213135*2^211088-1 is prime! (63553 decimal digits)  Time : 13.940 sec.
1005213135*2^211089-1 is not prime.  LLR Res64: 13CB4B3F3F4F8485  Time : 13.959 sec.
1012356645*2^211088-1 is prime! (63553 decimal digits)  Time : 14.069 sec.
1012356645*2^211089-1 is not prime.  LLR Res64: F22201FABF76C2B6  Time : 14.638 sec.
1020073659*2^211088-1 is prime! (63553 decimal digits)  Time : 14.305 sec.
1020073659*2^211089-1 is not prime.  LLR Res64: 9007354B82B61889  Time : 14.700 sec.
1033215159*2^211088-1 is prime! (63553 decimal digits)  Time : 13.426 sec.
1033215159*2^211089-1 is not prime.  LLR Res64: 6423ADD68D8C28E5  Time : 14.610 sec.
1035091365*2^211088-1 is prime! (63553 decimal digits)  Time : 14.399 sec.
1035091365*2^211089-1 is not prime.  LLR Res64: C49CA75CA4E8C3F1  Time : 13.995 sec.
1037115729*2^211088-1 is prime! (63553 decimal digits)  Time : 14.938 sec.
1037115729*2^211089-1 is not prime.  LLR Res64: CF44AB898D7409EC  Time : 14.778 sec.
1055050899*2^211088-1 is prime! (63553 decimal digits)  Time : 13.944 sec.
1055050899*2^211089-1 is not prime.  LLR Res64: B1126220835E8676  Time : 14.249 sec.
1060298607*2^211088-1 is prime! (63553 decimal digits)  Time : 14.511 sec.
1060298607*2^211089-1 is not prime.  LLR Res64: 88BFF56C221A7B40  Time : 13.985 sec.
1061175087*2^211088-1 is prime! (63553 decimal digits)  Time : 15.569 sec.
1061175087*2^211089-1 is not prime.  LLR Res64: 1B6CE013E5CDB077  Time : 14.362 sec.
1063734837*2^211088-1 is prime! (63553 decimal digits)  Time : 13.471 sec.
1063734837*2^211089-1 is not prime.  LLR Res64: 47207FD0D2CD8729  Time : 13.435 sec.
1067128107*2^211088-1 is prime! (63553 decimal digits)  Time : 13.059 sec.
1067128107*2^211089-1 is not prime.  LLR Res64: D5B514347C3EEA57  Time : 12.644 sec.
1068669447*2^211088-1 is prime! (63553 decimal digits)  Time : 12.540 sec.
1068669447*2^211089-1 is prime! (63554 decimal digits)  Time : 13.051 sec.

MooMoo2 · 2022-02-23, 05:44

Quote:

Originally Posted by MooMoo2

Testing's complete. k=1,068,669,447 is indeed the smallest k for which k*2^211088-1 and k*2^211089-1 are prime.

Well, I wondered what the lowest twin k was for n=211088, so I sieved for twins only this time. I checked that computer every 7-10 days or so, but nothing came up for a long time. I figured that 2/22/2022 is as good a day as any to check for twins, so I took a peek, and out came the following:

12566577633*2^211088-1 is prime! (63554 decimal digits) Time : 12.741 sec.
12566577633*2^211088+1 is prime! (63554 decimal digits) Time : 12.023 sec.

12599682117*2^211088-1 is prime! (63554 decimal digits) Time : 12.242 sec.
12599682117*2^211088+1 is prime! (63554 decimal digits) Time : 12.371 sec.

Yup, that's two twins found on 2/22/2022. Those are the 6th and 7th largest known twins.

I tried to submit them on 22:22:22, but as you can see - https://primes.utm.edu/primes/page.php?id=133622 I missed it by, you guessed it - two seconds. Oh well.

Stats:
Submitted: 2/22/2022 22:22:20 CDT
Range: k=1 - 12.71G
Candidates tested: ~3.899 million
n=211088 primes found: 1850 (including the twins)

2020-05-17, 23:56	#1
MooMoo2 "Michael Kwok" Mar 2006 1,181 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 k2^n+-1 and k2^(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 10686694472^211088-1 is prime! (63553 decimal digits) Time : 13.336 sec. 10686694472^211089-1 is prime! (63554 decimal digits) Time : 13.417 sec. 10686694472^211088+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 10686694472^211088-1 are 7,056,154,990,879,113...360,912,313,516,031, and the digits of 10686694472^211089-1 are 14,112,309,981,758,227...721,824,627,032,063. k=1,068,669,447 is likely the smallest k for which k2^211088-1 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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2020-05-18, 14:41	#4
ATH Einyen Dec 2003 Denmark 2²×863 Posts	Gratz on the top 10 Sophie Germain prime!

2020-05-18, 14:56	#5
Bottom Quark Dec 2010 31 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"

2020-05-19, 00:08	#7
VBCurtis "Curtis" Feb 2005 Riverside, CA 2×2,927 Posts	I have no idea what you are talking about.

2020-06-04, 09:41	#9
kar_bon Mar 2006 Germany 2·1,531 Posts	FYI: I've included this k-value into the Wiki and tested the whole range to n=250k.