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Hi,
below is a simple statistical calculation. PG is very close to the mean expected prime n: 8172825.9871397
1 0.0000000
2 2.0000000
3 1.5000000
4 1.3333333
6 1.5000000
7 1.1666667
11 1.5714286
18 1.6363636
34 1.8888889
38 1.1176471
43 1.1315789
55 1.2790698
64 1.1636364
76 1.1875000
94 1.2368421
103 1.0957447
143 1.3883495
206 1.4405594
216 1.0485437
306 1.4166667
324 1.0588235
391 1.2067901
458 1.1713555
470 1.0262009
827 1.7595745
1274 1.5405079
3276 2.5714286
4204 1.2832723
5134 1.2212179
7559 1.4723413
12676 1.6769414
14898 1.1752919
18123 1.2164720
18819 1.0384042
25690 1.3651097
26459 1.0299338
41628 1.5733021
51387 1.2344336
71783 1.3969097
80330 1.1190672
85687 1.0666874
88171 1.0289892
97063 1.1008495
123630 1.2737088
155930 1.2612634
164987 1.0580838
234760 1.4229000
414840 1.7670813
584995 1.4101702
702038 1.2000752
727699 1.0365522
992700 1.3641629
1201046 1.2098781
1232255 1.0259848
2312734 1.8768307
3136255 1.3560812
4235414 1.3504686
6090515 1.4379976
av 1.3418941
std.dev 0.2920128
std.error 0.0386780378
95% 1.2645379986 7701687.64868202
95% 1.4192501497 8643964.32559737
mean 8172825.9871397
Paul 



av 1.3418941
std.dev 0.2920128
std.error 0.0386780378
99.70% 2837301.30994368
95.00% 4615809.53567568
68.00% 6394317.76140769
average 8172825.9871397
68.00% 9087584.31966443
95.00% 11729842.4386037
99.70% 13508350.6643357
Gives a better picture of what is expected. But this is still a crude Gaussian normal curve for 3*2^n1. based on the last prime which was a little bigger than expected. If some one can do an expectation based on the Poisson distribution and post it here I am sure we would all be grateful, 


RogerVolunteer moderator Project administrator Volunteer developer Volunteer tester Project scientist
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To calculate a Poisson distribution you first find the average; 1.341894077
Poisson random variable must be an integer, so you need to divide the range into intervals. I used 0.1 but can be anything.
Then you have to find an average rate of success for each interval, starting with 1 to 1.1. Do to average what you did to range: (X1)*10
Using spreadsheet formula =POISSON(value,3.41894077) where value is 0,1,2,3.. corresponding to range 1 to 1.1, 1.1 to 1.2, 1.2 to 1.3,..
You end up with % chance to be in some range:
Range ; next n ; % chance
1.0 to 1.1 ; 6090515 to 6699567 ; 3.27%
1.1 to 1.2 ; 6699567 to 7308618 ; 11.20%
1.2 to 1.3 ; 7308618 to 7917670 ; 19.14%
1.3 to 1.4 ; 7917670 to 8526721 ; 21.81%
1.4 to 1.5 ; 8526721 to 9135773 ; 18.64%
1.5 to 1.6 ; 9135773 to 9744824 ; 12.75%
1.6 to 1.7 ; 9744824 to 10353876; 7.26%
1.7 to 1.8 ;10353876 to 10962927; 3.55%
1.8 to 1.9 ;10962927 to 11571979; 1.52%
1.9 to 2.0 ;11571979 to 12181030; 0.58%
2.0 to 2.1 ;12181030 to 12790082; 0.2%
2.1 to 2.2 ;12790082 to 13399133; 0.06%
2.2 to 2.3 ;13399133 to 14008185; 0.02%
____________



RogerVolunteer moderator Project administrator Volunteer developer Volunteer tester Project scientist
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I was a bit unsatified with the Poisson Distribution, so decided to graph the distribution by:
1. Sorting the ratio of prime finds. The list for k < 300 is given here: http://www.prothsearch.com/riesel2.html
2. Graphing the distribution by giving each ratio an equal weight. This is the blue line in the graph:
3. Finding line of best fit. That is the red line in the above graph. The equation I got was:
132.122*e^(3.395*X), where X = last prime * n
This result is satisfying because the cumulative distribution of the gaps between the values of a Poisson process should be governed by an exponential distribution:
http://primes.utm.edu/notes/faq/NextMersenne.html
Running the numbers:
Range ; next n ; % chance
1.0 to 1.1 ; 6090515 to 6699567 ; 31.02%
1.1 to 1.2 ; 6699567 to 7308618 ; 22.09%
1.2 to 1.3 ; 7308618 to 7917670 ; 15.73%
1.3 to 1.4 ; 7917670 to 8526721 ; 11.20%
1.4 to 1.5 ; 8526721 to 9135773 ; 7.98%
1.5 to 1.6 ; 9135773 to 9744824 ; 5.68%
1.6 to 1.7 ; 9744824 to 10353876; 4.05%
1.7 to 1.8 ;10353876 to 10962927; 2.88%
1.8 to 1.9 ;10962927 to 11571979; 2.05%
1.9 to 2.0 ;11571979 to 12181030; 1.46%
2.0 to 2.1 ;12181030 to 12790082; 1.04%
2.1 to 2.2 ;12790082 to 13399133; 0.74%
2.2 to 2.3 ;13399133 to 14008185; 0.53%
2.3 to inf ;greater than 14008185; 1.31%
You can see the new distribution is much thicker at the start and in the tail. I plan to also try this out for other k.
Bottom line is crunch early, crunch often!
____________



HonzaVolunteer moderator Volunteer tester Project scientist Send message
Joined: 15 Aug 05 Posts: 1732 ID: 352 Credit: 2,144,072,153 RAC: 1,270,922

With n ~9,37m, aren't we a bit overdue with next 321 prime?
3·2^7033641+1 (321) 2011/02
3·2^60905151 (321) 2010
3·2^5082306+1 (321) 2009
3·2^42354141 (321) 2008
I was thinking to swich to longer subproject during server move/outage.
321 might not be long enough for fast computers but still quite appealing.
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Badge score: 1*1 + 5*2 + 8*9 + 9*6 + 12*3 = 173 



Thanks for your statistics, Roger.
Remember that this is for half of 321 Prime Search. Not only are we due a prime of the form 3*2^n1, there is also the chance of finding a prime of the form 3*2^n+1. PrimeGrid is at about 9,369,596 bits. All the more reason to crunch for this subproject  a top20 prime is nigh! 


RogerVolunteer moderator Project administrator Volunteer developer Volunteer tester Project scientist
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Doing the exercise for the 3*2^n+1 case gives equation of the form 15.152*e^(1.630*X), where X = last prime * n
Running the numbers:
Range ; next n ; % chance
1.0 to 1.1 ; 7033641 to 7737005 ; 14.29%
1.1 to 1.2 ; 7737005 to 8440369 ; 12.90%
1.2 to 1.3 ; 8440369 to 9143733 ; 10.96%
1.3 to 1.4 ; 9143733 to 9847097 ; 9.31%
1.4 to 1.5 ; 9847097 to 10550462; 7.91%
1.5 to 1.6 ;10550462 to 11253826; 6.72%
1.6 to 1.7 ;11253826 to 11957190; 5.71%
1.7 to 1.8 ;11957190 to 12660554; 4.85%
1.8 to 1.9 ;12660554 to 13363918; 4.12%
1.9 to 2.0 ;13363918 to 14067282; 3.50%
2.0 to 2.1 ;14067282 to 14770646; 2.97%
2.1 to 2.2 ;14770646 to 15474010; 2.53%
2.2 to 2.3 ;15474010 to 16177374; 2.15%
2.3 to inf ;greater than 16177374;12.12%
When we add the 3*2^n+1 and 3*2^n1 probabilities for up and coming ranges we get:
Range ; % chance
9.37 to 9.4M ; 0.5%
9.4 to 9.6M ; 4.46%
9.6 to 9.8M ; 4.15%
9.8 to 10M ; 3.87%
10 to 10.5M ; 8.57%
10.5 to 11M ; 7.24%
11 to 11.5M ; 6.14%
11.5 to 12M ; 5.24%
12 to 12.5M ; 4.49%
12.5 to 13M ; 3.87%
13 to 13.5M ; 3.35%
13.5 to 14M ; 2.90%
14 to 14.5M ; 2.53%
14.5 to 15M ; 2.21%
15 to 15.5M ; 1.93%
15.5M to inf ; 14.78%
Note that these probabilities don't take account of the fact prime was not found < 9.37M.
When we normailise (by making the remaining range add up to 100% chance for both +/ cases) we get:
Range ; next n ; % chance
9.37 to 9.4M ; 1.67%
9.4 to 9.6M ; 14.83%
...
All numbers to 25M are loaded in the Boinc Database. There is a table with only 321 work up to 25M.
If someone could run a Database query for how many n there are to be checked in the up and coming ranges we could see the 1 in X chance for each WU of finding a huge prime!
____________



RogerVolunteer moderator Project administrator Volunteer developer Volunteer tester Project scientist
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Thanks to JimB for querying the database for the number of raw candidates in the up and coming ranges!
Adding the 3*2^n+1 and 3*2^n1 normalised probabilities (so search from here adds up to 100% chance for each +/):
Range ; candidates ; % chance ; 1 in X chance of prime per candidate
9.4 to 9.6M ; 17462 ; 14.98% ; 116,601
9.6 to 9.8M ; 17528 ; 13.68% ; 128,133
9.80 to 10M ; 17402 ; 12.51% ; 139,144
10 to 10.5M ; 43550 ; 26.84% ; 162,287
10.5 to 11M ; 43627 ; 21.62% ; 201,790
11 to 11.5M ; 43365 ; 17.52% ; 247,450
11.5 to 12M ; 43468 ; 14.29% ; 304,089
12 to 12.5M ; 43590 ; 11.73% ; 371,481
12.5 to 13M ; 43264 ; 9.69% ; 446,296
13 to 13.5M ; 43761 ; 8.06% ; 542,982
13.5 to 14M ; 43476 ; 6.74% ; 644,860
14 to 14.5M ; 43775 ; 5.67% ; 771,553
14.5 to 15M ; 43387 ; 4.80% ; 903,541
15 to 15.5M ; 43695 ; 4.09% ; 1,069,381
15.5M to inf ; 827810 (to 25M); 27.77% ; 2,980,636
Considering that a prime at the current n would be in the top 15 ever found, you should consider 321 while these candidates have such good chances!
____________



HonzaVolunteer moderator Volunteer tester Project scientist Send message
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Thanks Roger, this is somehow encouraging :)
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Badge score: 1*1 + 5*2 + 8*9 + 9*6 + 12*3 = 173 


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Range ; candidates ; % chance ; 1 in X chance of prime per candidate
9.4 to 9.6M ; 17462 ; 14.98% ; 116,601
9.6 to 9.8M ; 17528 ; 13.68% ; 128,133
9.80 to 10M ; 17402 ; 12.51% ; 139,144
10 to 10.5M ; 43550 ; 26.84% ; 162,287
10.5 to 11M ; 43627 ; 21.62% ; 201,790
11 to 11.5M ; 43365 ; 17.52% ; 247,450
11.5 to 12M ; 43468 ; 14.29% ; 304,089
12 to 12.5M ; 43590 ; 11.73% ; 371,481
12.5 to 13M ; 43264 ; 9.69% ; 446,296
13 to 13.5M ; 43761 ; 8.06% ; 542,982
13.5 to 14M ; 43476 ; 6.74% ; 644,860
14 to 14.5M ; 43775 ; 5.67% ; 771,553
14.5 to 15M ; 43387 ; 4.80% ; 903,541
15 to 15.5M ; 43695 ; 4.09% ; 1,069,381
15.5M to inf ; 827810 (to 25M); 27.77% ; 2,980,636
Am I reading this correct? Are you saying that a candidate in 9.49.6 range is 10 times more likely to be a prime than one in the 1515.5 range? I don't think that is correct. 



Range ; candidates ; % chance ; 1 in X chance of prime per candidate
9.4 to 9.6M ; 17462 ; 14.98% ; 116,601
9.6 to 9.8M ; 17528 ; 13.68% ; 128,133
9.80 to 10M ; 17402 ; 12.51% ; 139,144
10 to 10.5M ; 43550 ; 26.84% ; 162,287
10.5 to 11M ; 43627 ; 21.62% ; 201,790
11 to 11.5M ; 43365 ; 17.52% ; 247,450
11.5 to 12M ; 43468 ; 14.29% ; 304,089
12 to 12.5M ; 43590 ; 11.73% ; 371,481
12.5 to 13M ; 43264 ; 9.69% ; 446,296
13 to 13.5M ; 43761 ; 8.06% ; 542,982
13.5 to 14M ; 43476 ; 6.74% ; 644,860
14 to 14.5M ; 43775 ; 5.67% ; 771,553
14.5 to 15M ; 43387 ; 4.80% ; 903,541
15 to 15.5M ; 43695 ; 4.09% ; 1,069,381
15.5M to inf ; 827810 (to 25M); 27.77% ; 2,980,636
Am I reading this correct? Are you saying that a candidate in 9.49.6 range is 10 times more likely to be a prime than one in the 1515.5 range? I don't think that is correct.
Intuitively that sounds right to me since as numbers get larger, the probability of a number being prime goes down by quite a bit (i.e.: goes down more than just linearly).
____________



axnVolunteer developer Send message
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Intuitively that sounds right to me since as numbers get larger, the probability of a number being prime goes down by quite a bit (i.e.: goes down more than just linearly).
Nope. It goes down precisely linearly. This phenomenon is called, surprisingly, "Prime Number Theorem" :) 


RogerVolunteer moderator Project administrator Volunteer developer Volunteer tester Project scientist
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Prime Number Theory say that the chance of a random integer x being prime is about 1/log x
http://primes.utm.edu/howmany.shtml#3
For x = 3*2^n+/1, chance is about 1/log (3*2^n+/1).
As n gets large this chance approaches 1/n.
____________




Hello!
Edit :: My bad, it's already reported in this thread:
http://www.primegrid.com/forum_thread.php?id=5462
At least it is now official on the list!
If I'm not wrong, another megaprime is found!
New entry at 12'th place:
http://primes.utm.edu/primes/page.php?id=116922
Seems like 2014 will be a very nice year primewise!!
With regards,
Hans Sveen
Oslo, Norway
Keep on searching!!
What the heck, what is JStriders signature doing here??[/url]
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[DPC]CharleyVolunteer moderator Project scientist Send message
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Hello1
If I'm not wrong, another megaprime is found!
New entry at 12'th place:
http://primes.utm.edu/primes/page.php?id=116922
Seems like 2014 will be a very nice year primewise!!
With regards,
Hans Sveen
Oslo, Norway
Keep on searching!!
What the heck, what is JStriders signature doing here??!
Yep, that's one of the mega primes mentioned in this thread[/]. As is also posted there I believe, there hasn't been an official announcement here because there are still some tests being run :)
____________
[url=http://www.charleygielkens.nl/pg]PrimeGrid Challenge Overall standings  Last update: From Pi to Paddy (2016)




The last "" 321 prime is: 3*2^60905151 1833429 L1353 2010. Will the 2n rule be broken? I.e. to date there has always been at least on prime less than exponent 2n where there has been a prime for exponent n. So, given PG is crunching at 10862313, PG should find a "" 321 prime real soon, fingers crossed, toes crossed, rabbit's foot worn,... :) 



I am crunching 321 on an old (4 core) Q6600 now, with a reasonable chance of finding a huge prime. Tests are taking 60 hours per core  that is about 11 tests per week for the box. I pleased to say that PG on BOINC works seamlessly. 



Last night, my old Q6600 blew up with a bang! followed by a great stink. I stripped down the box and in the process broke the mounting bracket for the Arctic Freezer 7. I now have some spare parts! I put my old faithful Tyan 4x4 AMD CPU onto 321. I will get about 3035 tests per week out of it. 


DaveSend message
Joined: 13 Feb 12 Posts: 2208 ID: 130544 Credit: 613,352,997 RAC: 137,099

Once I've topped out the final few SGS, helped a bit with TRP cleanup & got my final 10 WOO in the bag, I'll be joining you for 1000000cr/(25k/dy)=40 days. 



Well I have been running the quad socket Tyan for 10 days and have these stats:
State: All 49  In progress 16  Pending 1  Inconclusive 13  Valid 7  Aborted  Invalid 12  Error
I think I have fixed the issues with it. Mostly turning "throttle" "enabled"  for some reason the BIOS had reset itself after a long period of no use; and active cooling of the HT heatspeaders.
There is some consolation: I have my 321 gold badge! 


DaveSend message
Joined: 13 Feb 12 Posts: 2208 ID: 130544 Credit: 613,352,997 RAC: 137,099

Well done :).
After the BIOS reset, did you load optimised defaults or a similar option? 



Well done :).
After the BIOS reset, did you load optimised defaults or a similar option?
No. I just poked around. The OS was to windows  I am Linux. I had to tweak which graphics card to use. Turned off floppy. Made IMMOU 1GB. Turn on throttle. Turn on PowerNow. And a few other non performance related things.
I really hope it is 100% for 321 now. I will give another week and if there is no improvement I will have to think again. 


Michael GoetzVolunteer moderator Project administrator Project scientist
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Typically that problem is caused by overclocking.
____________
Please do not PM me with support questions. They will usually go unanswered. Ask on the forums instead. Thank you!
My lucky number is 75898^524288+1 



Typically that problem is caused by overclocking.
The Tyan has no overclocking. I think my problem was the throttle and hot HTs. I should get the first results through with the new settings in a couple of days
I hope others will not overclock! A 5% speed up is not worth it if 1 in 10 results are bad. 



As far as I can tell, 4 day 321 tests seems to continue to turn in invalid results on the Tyan. I have moved it over to SGS and will see how it gets on there. 


Michael GoetzVolunteer moderator Project administrator Project scientist
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As far as I can tell, 4 day 321 tests seems to continue to turn in invalid results on the Tyan. I have moved it over to SGS and will see how it gets on there.
You'll see a higher percentage of successful tasks, but that's only because the much shorter tasks are more likely to finish before an error occurs. The problem won't go away. If anything, shorter tasks are actually tougher on the CPU because of the smaller FFT sizes.
If you're not overclocking, the next most common problem is bad memory. Sometimes this can be corrected by removing and reseating the memory cards.
Next after that is overheating. If the heatsinks and fans are clogged with dust, or coated with smoke residue from either cooking or tobacco, the CPU or motherboard components could be overheating and malfunctioning.
Also try disabling antivirus programs, or excluding them from the BOINC directories. (Note: with AVG you need to completely disable it's "identity protection" system. It can't be excluded from individual directories and it DOES cause problems with BOINC.)
I've never heard of a throttling problem causing errors, but I suppose anything is possible.
____________
Please do not PM me with support questions. They will usually go unanswered. Ask on the forums instead. Thank you!
My lucky number is 75898^524288+1 



1 0.0000000
2 2.0000000
3 1.5000000
4 1.3333333
6 1.5000000
7 1.1666667
11 1.5714286
18 1.6363636
34 1.8888889
38 1.1176471
43 1.1315789
55 1.2790698
64 1.1636364
76 1.1875000
94 1.2368421
103 1.0957447
143 1.3883495
206 1.4405594
216 1.0485437
306 1.4166667
324 1.0588235
391 1.2067901
458 1.1713555
470 1.0262009
827 1.7595745
1274 1.5405079
3276 2.5714286
4204 1.2832723
5134 1.2212179
7559 1.4723413
12676 1.6769414
14898 1.1752919
18123 1.2164720
18819 1.0384042
25690 1.3651097
26459 1.0299338
41628 1.5733021
51387 1.2344336
71783 1.3969097
80330 1.1190672
85687 1.0666874
88171 1.0289892
97063 1.1008495
123630 1.2737088
155930 1.2612634
164987 1.0580838
234760 1.4229000
414840 1.7670813
584995 1.4101702
702038 1.2000752
727699 1.0365522
992700 1.3641629
1201046 1.2098781
1232255 1.0259848
2312734 1.8768307
3136255 1.3560812
4235414 1.3504686
6090515 1.4379976
11484018 1.8855578
The average increase, over the previous, is 1.3512676. So it seems to me the next 321 (minus form) will come very soon.
[/code] 



What I'd like to see would be a 3*2^n +/ 1 twin prime in the range we're searching now. Wouldn't that be an event! I know, I know, "don't hold your breath". :)
Gary 



What I'd like to see would be a 3*2^n +/ 1 twin prime in the range we're searching now. Wouldn't that be an event! I know, I know, "don't hold your breath". :)
Gary
Also see OEIS A181490. /JeppeSN 



1 0.0000000
2 2.0000000
3 1.5000000
4 1.3333333
6 1.5000000
7 1.1666667
11 1.5714286
18 1.6363636
34 1.8888889
38 1.1176471
43 1.1315789
55 1.2790698
64 1.1636364
76 1.1875000
94 1.2368421
103 1.0957447
143 1.3883495
206 1.4405594
216 1.0485437
306 1.4166667
324 1.0588235
391 1.2067901
458 1.1713555
470 1.0262009
827 1.7595745
1274 1.5405079
3276 2.5714286
4204 1.2832723
5134 1.2212179
7559 1.4723413
12676 1.6769414
14898 1.1752919
18123 1.2164720
18819 1.0384042
25690 1.3651097
26459 1.0299338
41628 1.5733021
51387 1.2344336
71783 1.3969097
80330 1.1190672
85687 1.0666874
88171 1.0289892
97063 1.1008495
123630 1.2737088
155930 1.2612634
164987 1.0580838
234760 1.4229000
414840 1.7670813
584995 1.4101702
702038 1.2000752
727699 1.0365522
992700 1.3641629
1201046 1.2098781
1232255 1.0259848
2312734 1.8768307
3136255 1.3560812
4235414 1.3504686
6090515 1.4379976
11484018 1.8855578
11731850 1.0215806
The average increase is now 1.3456797, so just maybe we will not have to wait very long for the next 321 (minus form) prime. Of course, in the mean time a plus form might be found ;)




1 0.0000000
2 2.0000000
3 1.5000000
4 1.3333333
6 1.5000000
7 1.1666667
11 1.5714286
18 1.6363636
34 1.8888889
38 1.1176471
43 1.1315789
55 1.2790698
64 1.1636364
76 1.1875000
94 1.2368421
103 1.0957447
143 1.3883495
206 1.4405594
216 1.0485437
306 1.4166667
324 1.0588235
391 1.2067901
458 1.1713555
470 1.0262009
827 1.7595745
1274 1.5405079
3276 2.5714286
4204 1.2832723
5134 1.2212179
7559 1.4723413
12676 1.6769414
14898 1.1752919
18123 1.2164720
18819 1.0384042
25690 1.3651097
26459 1.0299338
41628 1.5733021
51387 1.2344336
71783 1.3969097
80330 1.1190672
85687 1.0666874
88171 1.0289892
97063 1.1008495
123630 1.2737088
155930 1.2612634
164987 1.0580838
234760 1.4229000
414840 1.7670813
584995 1.4101702
702038 1.2000752
727699 1.0365522
992700 1.3641629
1201046 1.2098781
1232255 1.0259848
2312734 1.8768307
3136255 1.3560812
4235414 1.3504686
6090515 1.4379976
11484018 1.8855578
11731850 1.0215806
11895718 1.0139678
The average is now 1.3401511. Can we expect a fourth bus to come along? ;)




99.70% 1.22428235452968 14059680.5965012
95.00% 1.26290528305169 14503227.0028607
68.00% 1.30152821157371 14946773.4092203
average 1.3401511 15390319.8155799
68.00% 1.37877406861775 15833866.2219395
95.00% 1.41739699713977 16277412.628299
99.70% 1.45601992566179 16720959.0346586
Is my calculation for next 3*2^n1 prime.
Today the Year of the Dog challenge passed the 14M bit mark.
There is also a 3*2^n+1 prime out there somewhere. 



I did a fun calculation based on the density distribution of the ratios and their local peaks using a kernel density estimation.
Basically I just look at around which ratios "clump" together and assume that there would be some systematic behaviour in it (which there most likely is not). The statistics are also quite low. I excluded a few of the first ratios.
I predict that 321*n1 will be prime at
ratio, chance, n
1.34 41.46 % 15940262
1.566 15.65 % 18628694
1.65 9.79 % 19627935
1.762 11.20 % 20960255
1.886 16.37 % 22435324
2.571 5.53 % 30583891
There is also a local "clumps" at 1.042 and 1.209, but we are already past these.
I predict that 321*n+1 will be prime at
ratio, chance, n
1.342 25.82 % 14532982
1.541 9.33 % 16688022
1.626 14.71 % 17608517
1.854 9.10 % 20077607
1.987 13.63 % 21517911
2.864 9.14 % 31015247
4.135 9.14 % 44779346
5.345 9.14 % 57882854
There is also a local "clumps" at 1.067, 1.139 and 1.22, but we are already past these.
These numbers are based on the local peaks of the distribution, so the error estimates are very large!
But I would say that there is a possibility that a 321*n+1 prime is close :) 



Do you write 321*n±1 to mean 3*2^n±1?
They are not the same. While 321*n±1 is known to be prime infinitely often (each arithmetic progression a*n+1 or a*n1 contains infinitely many primes), for almost all really big values of n there is no efficient way to check if 321*n±1 is prime.
The numbers 3*2^n±1 are much more scarce, but for each of them there is an efficient way to check its primality.
/JeppeSN 



Ugh! :D
I definately mean 3*2^n1 and 3*2^n+1!
Just a brain fart on my part :) 



Also: based on my tests, the FFT size will increase from 768K to 800K at n=14536766. 



Also: based on my tests, the FFT size will increase from 768K to 800K at n=14536766.
By my calculations, there is about 11001200 candidates left before the rise in FFT size, which is about a 4.2% increase. 

