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Old 2019-07-26, 20:51   #1
jaydfox
 
Jul 2019

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Default Where to find Prime Gap lists?

Hi Everyone,

I apologize for posting out of the blue. I've skimmed through the forum, and I've read through some of the threads, but I wasn't really sure where to find an easily accessible list of known prime gaps. I don't mean a random large gap here or there. I mean, is there a database or list of ALL the "large" prime gaps above a certain size below a certain limit?

For example, on Dr. Nicely's website, I found a list of all the "kilogaps" below 5e16. Actually, it was a list of all known first occurrences of gaps, and a separate list of all the kilogaps below 5e16 that weren't already in the first occurrence list. I had to merge the two lists. But the point stands, this is presumably a list of all the gaps of at least 1000, below 5e16. (Incidentally, I counted 149 such gaps after merging the two lists. I have no way of verifying that number.)

But given that the maximal gaps have been proven up to at least 1.8e19, I am hoping that somewhere, someone is maintaining a list of all the kilogaps below 1e19. Actually, I assume Dr. Nicely would have access to this data, but I spent some time going through his website, and I couldn't seem to figure out how to get the data I'm looking for.

I realize that there will be several thousand such gaps, but in the grand scheme of things, it seems like a very small amount of data to maintain. Does anyone know where to find it? Is it already listed publicly somewhere? Is there a private list that is accessible to people who are actively participating in these prime gap searches? I'm not necessarily at the point of wanting to throw my spare processor cycles at such a search. I'm still trying to understand the theory and run experiments, and I'm looking for high quality data to verify/challenge my understanding.
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Old 2019-07-27, 09:31   #2
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Quote:
Originally Posted by jaydfox View Post
Hi Everyone,

I apologize for posting out of the blue. I've skimmed through the forum, and I've read through some of the threads, but I wasn't really sure where to find an easily accessible list of known prime gaps. I don't mean a random large gap here or there. I mean, is there a database or list of ALL the "large" prime gaps above a certain size below a certain limit?

For example, on Dr. Nicely's website, I found a list of all the "kilogaps" below 5e16. Actually, it was a list of all known first occurrences of gaps, and a separate list of all the kilogaps below 5e16 that weren't already in the first occurrence list. I had to merge the two lists. But the point stands, this is presumably a list of all the gaps of at least 1000, below 5e16. (Incidentally, I counted 149 such gaps after merging the two lists. I have no way of verifying that number.)

But given that the maximal gaps have been proven up to at least 1.8e19, I am hoping that somewhere, someone is maintaining a list of all the kilogaps below 1e19. Actually, I assume Dr. Nicely would have access to this data, but I spent some time going through his website, and I couldn't seem to figure out how to get the data I'm looking for.

I realize that there will be several thousand such gaps, but in the grand scheme of things, it seems like a very small amount of data to maintain. Does anyone know where to find it? Is it already listed publicly somewhere? Is there a private list that is accessible to people who are actively participating in these prime gap searches? I'm not necessarily at the point of wanting to throw my spare processor cycles at such a search. I'm still trying to understand the theory and run experiments, and I'm looking for high quality data to verify/challenge my understanding.

Hi jaydfox

Originally we had the idea that we would collect all kilogaps in the range 4e18 to 2^64, but in fact it was not done - we opted for faster programme settings, which reported on gaps of 1000 or more, but which did not produce a definitive list of all kilogaps. Given that this program is still available at this site, you could reduce the variables to report all kilogaps, but this would run a bit slower than the settings we used, and would need to be run from 5e16 upwards rather than 4e18. The work we carried out used about 5-10 high performance desktops, and took over a year. So producing the definitive kilogap list to 2^64 is not an insignificant task.
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Old 2019-07-27, 18:43   #3
jaydfox
 
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Quote:
Originally Posted by robert44444uk View Post
Hi jaydfox

Originally we had the idea that we would collect all kilogaps in the range 4e18 to 2^64, but in fact it was not done - we opted for faster programme settings, which reported on gaps of 1000 or more, but which did not produce a definitive list of all kilogaps. Given that this program is still available at this site, you could reduce the variables to report all kilogaps, but this would run a bit slower than the settings we used, and would need to be run from 5e16 upwards rather than 4e18. The work we carried out used about 5-10 high performance desktops, and took over a year. So producing the definitive kilogap list to 2^64 is not an insignificant task.
Thanks for the reply! I have a six-year-old laptop, rather modest in performance, so I doubt I could add much to the overall search for maximal gaps. But I'll download the code and play around with it, maybe see if I can help extend the list of kilogaps above 5e16.
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Old 2019-07-30, 19:51   #4
mart_r
 
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Quote:
Originally Posted by jaydfox View Post
Thanks for the reply! I have a six-year-old laptop, rather modest in performance, so I doubt I could add much to the overall search for maximal gaps. But I'll download the code and play around with it, maybe see if I can help extend the list of kilogaps above 5e16.

Two years ago I searched a small range for kilogaps >5e16, because I wanted to find eight more gaps of length 1000 so I'd have a list of the first 20 of them. Didn't find any up to 5.2e16, but if you're interested I could do some more searching there.

Here are results from 5e16 to 5.2e16, all gaps >= 1000:
Code:
1080 50393998281635599
1036 50597782557370831
1038 50662912533154139
1028 50669505541630601
1020 51103842778483477
1006 51178110554293441
1036 51506543650015963
1022 51540068502364151
1030 51821938103395729
1028 51878675840390051
1024 51922353423303283
BTW, aren't you also in the tetration forum?

Last fiddled with by mart_r on 2019-07-30 at 19:54 Reason: grammar
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Old 2019-07-31, 00:12   #5
jaydfox
 
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Quote:
Originally Posted by mart_r View Post
Two years ago I searched a small range for kilogaps >5e16, because I wanted to find eight more gaps of length 1000 so I'd have a list of the first 20 of them. Didn't find any up to 5.2e16, but if you're interested I could do some more searching there.

Here are results from 5e16 to 5.2e16, all gaps >= 1000:
Hey, thanks for that! I've downloaded the gap12.c code, and I've been reading through it, but I haven't tried compiling it yet. I'll add these gaps to the list I scraped from Dr. Nicely's site.

I've been trying to do some analysis of the modified Cramér model (after sieving small primes), but I lack quality data for testing the model. I need gaps that are large enough (relative to the average gap) to test the suitability of the model to estimate maximal gaps, but I need enough gaps to actually get statistically significant results. Hence my interest in kilogaps.

I actually think a more interesting set of data to test the model is the nth largest gap (i.e., the nth largest gap seen so far, with a tie considered "smaller" than a previous occurrence of the same gap). The data get smoother as n is increased, so it's easier to see the trends in the data. Having access to a complete list of kilogaps (>= 1000) would facilitate this, since I could start analyzing from the nth kilogap.

Quote:
Originally Posted by mart_r View Post
BTW, aren't you also in the tetration forum?
Yes, I've been active off and on in that forum for... I guess over a dozen years now. But I've taken some long breaks. Family life, work, etc., gets in the way of my math hobby sometimes.
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Old 2019-07-31, 19:06   #6
mart_r
 
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One of my PCs is still dedicated to the twin prime gaps search (probably another week or two, there was a slight delay due to a little heatwave here in Germany last week with up to 41°C, and I paused the calculation for a few days).

With the other I can currently search a range of about 7e14 per day, which is not much, but I'll continue unless you (or someone else) have significantly more resources.
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Old 2019-08-27, 18:35   #7
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Kilogaps between 5e16 and 6.1e16:
Code:
1080 50393998281635599
1036 50597782557370831
1038 50662912533154139
1028 50669505541630601
1020 51103842778483477
1006 51178110554293441
1036 51506543650015963
1022 51540068502364151
1030 51821938103395729
1028 51878675840390051
1024 51922353423303283
1068 52604763098088133
1040 52888336638766061
1012 52924388177774671
1004 52940197762336457
1038 53190502026869531
1030 53424413597601847
1098 53638706154728053
1012 53660015663845999
1002 53707322644803361
1020 53914110235891273
1010 54014460400754531
1016 54238958822744543
1074 54406581745449079
1036 54440388168355723
1020 54882837712797953
1028 55113259135704059
1004 55163718630690983
1004 55200837956308043
1002 55283761083282649
1198 55350776431903243
1058 55449498692096303
1008 55487720105088083
1020 55488741963678451
1056 55682178862196023
1050 55732090121186321
1020 55823579213985769
1020 55891416733516277
1018 56091766320454369
1000 56136441023068027
1034 56158582731193853
1000 56670104952987829
1062 56981519303622899
1050 57155528198687827
1024 57222906071591899
1040 57654402011303759
1128 57701567767397551
1032 57753877128029321
1050 57776087064021863
1008 58194865424950141
1068 58555411853875691
1018 58598010042460591
1028 58616959313839973
1020 58819143208929863
1010 58838200694113361
1160 58904398834827743
1014 58962322496980073
1018 59040706748307103
1026 59196144582231763
1030 59407829528206891
1026 59490062497256117
1020 59541933150765029
1032 59589625441970119
1050 59818063210518643
1134 59847994456941689
1050 59871385309648697
1032 59914213666020719
1038 60133080805428509
1010 60502367225643683
1022 60553417610117651
1146 60592199607166471
1056 60638802299711093
1022 60765773554580597
1016 60802458897362753
1008 60896327833261201
Still doing some searching, but not with the highest priority.
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Old 2019-08-29, 00:50   #8
jaydfox
 
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Quote:
Originally Posted by mart_r View Post
Kilogaps between 5e16 and 6.1e16:

Still doing some searching, but not with the highest priority.
Hey, wow, thanks for this! That's 75 more to add to my list of 149. I.e., 149 kilogaps under 5e16, and 224 total under 6.1e16.

Of the 75 in your list, I had 6 from Dr. Nicely's website, so that added 69 more kilogaps!
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Old 2019-10-24, 21:25   #9
mart_r
 
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Some more kilogaps between 6.1e16 and 7.4e16:

Code:
1000 61289814801068713
1166 61499427052410467
1022 61608029101269257
1026 61707218844150821
1002 61852389306473867
1032 62045478361471351
1002 62269241894354171
1000 62361936121863601
1040 62808550411710269
1008 63030750779743111
1068 63585518075296193
1076 63751388393794781
1002 64025048118749989
1022 64064698065478949
1046 64178237253890621
1022 64435216702531091
1004 64437453352197017
1010 64489035614704343
1038 64656379256421941
1072 64681887054028429
1040 64903409232167393
1018 65171327701466443
1016 65303784137621093
1008 65541152120090963
1004 65656259060122589
1082 65850655232778077
1018 65865648969941053
1050 66282988287629461
1010 66295898802923987
1008 66316366986016003
1122 66322416129057029
1012 66389213712846301
1004 66507821823362993
1020 66677148675716687
1064 66712486906360703
1008 66803921756273941
1002 66820739930230501
1014 67033743226720297
1002 67306562390173597
1014 67419425658776777
1004 67631621153942789
1032 67646010960504871
1006 67699163242627531
1174 67777708053772723
1128 68248526808384701
1090 68402499016330903
1058 68428465104182033
1024 68475318115749673
1008 68483650739629969
1076 68607677530724921
1028 68680143205806719
1020 68858613651275563
1006 68889111160953541
1002 69096269699560111
1002 69146992679052011
1050 69363019583598199
1038 69414990756773269
1014 69603276634606153
1000 69672350035847929
1022 69707999361413111
1022 69887880784634327
1050 70072593169404983
1014 70133104083953173
1012 70207452984829309
1020 70514057617500059
1080 70682835348242081
1038 71291937976937471
1002 71334390972787619
1006 71732350211993017
1040 71912635508975399
1060 72091252389474481
1020 72119924546819353
1032 72245286734288309
1078 72638824545480979
1014 72660209757895937
1012 72714085593695191
1034 73081138652291327
1006 73132086814502323
1030 73280655706629409
1026 73320383505423721
1036 73342849465603087
1002 73456185314120771
1040 73606458765637037
1008 73960629782222201
Maybe one or two entries before 7.3e16 are still missing, as that range is not yet fully processed (spare CPU time on my laptop which is ... slow).
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Old 2019-12-14, 16:55   #10
mart_r
 
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Now that I found my 20 gaps with g=1000 and stopping this search, here are the kilogaps for 74e15 to 81e15, including one I was hinting at in my previous post:

Code:
1008 72785657893044751

1158 74224828152967109
1008 74253247947930853
1014 74427355957362047
1068 74566933953299951
1050 74672144302283221
1040 74732186510943107
1036 74949586587024841
1020 74986031869324259
1010 75120665420746919
1020 75781977412958353
1008 75925671745922261
1034 76082709975153047
1002 76087968329099347
1038 76357211355690061
1074 76485516106246547
1050 76641652530262307
1020 76706609121161687
1002 77033734179518617
1062 77099714753631979
1004 77481882268446659
1000 77564216289320659
1000 77671773242528989
1022 77702137670610647
1008 77744782220928523
1016 77787200404407383
1014 78038973615245437
1092 78120687162386527
1132 78137034924137419
1118 78661510510855829
1010 78781883279456333
1156 78944802602538877
1030 78993860918645443
1140 79017505670475503
1038 79140537078226669
1072 79525408741854637
1038 79647230167781779
1032 79714152290449507
1012 79836836014438207
1044 80285560147848283
1036 80416867635858673
1056 80444074990611217
1132 80464625759765719
1014 80564519093188279
1002 80597718041848247
1034 80630172261943979
1002 80806550204385689
1220 80873624627234849
1018 80909422108812589
1000 81065679461847721
1030 81171954964536067
1018 81264833949895861
1014 81269972698996723
1020 81330531155678237
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Old 2019-12-17, 17:48   #11
storm5510
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Fascinating! This was something I experimented with many years ago. I wrote a few programs. The problem was that I did not have access to a big integer library. Now, I do. So, I revisited it, adding the big integer capability. I do not know what the ceiling of this library is, but it is way up there. I had it display the decimal digits of 2^1619-1. It filled the screen.

It is sort of hard to tell what all of you are searching for by looking at your lists. Smallest gap, largest gap, I am not sure. Either way, I cannot run my little program flat-out. The accumulated data would be overwhelming to pick through. I choked it down as much as I could. Below is a sample of the output

Code:
12-17-2019 12:30:42: 14739, 160669 - 160663 = 6
12-17-2019 12:30:42: 14740, 160681 - 160669 = 12
12-17-2019 12:30:42: 14741, 160687 - 160681 = 6
12-17-2019 12:30:42: 14742, 160697 - 160687 = 10
12-17-2019 12:30:42: 14743, 160709 - 160697 = 12
12-17-2019 12:30:42: 14744, 160711 - 160709 = 2
It needs a filter for the output data. What this would be, again, I am not sure.
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