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Prime k-tuplets


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Abstract

At this site I have collected together all the largest known examples of certain types of dense clusters of prime numbers. The idea is to generalise the notion of prime twins - pairs of prime numbers {p, p + 2} - to groups of three or more.

Prepared by Tony Forbes (1997- Aug 2021); anthony.d.forbes@gmail.com.

Old site addresses: http://www.ltkz.demon.co.uk/ktuplets.htm.
http://anthony.d.forbes.googlepages.com/ktuplets.htm

Continued by Norman Luhn.
Contact: pzktupel[at]pzktupel[dot]de

This site address: https://pzktupel.de/ktuplets.php

Additions ( last updated: 02 July 2022 )

Recent additions

History of this web site → 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021

Contents

  1. Introduction
  2. The Largest Known Prime k-tuplets
  3. Summary
  4. Mathematical Background
  5. References
  6. Collection of Prime Page Links
  7. If you like, you can help to save this huge work for future - please download and store it ! ( Filesize ~2.7 MB, 28 Jun 2022 )

1. Introduction

Prime Numbers

Prime numbers are the building blocks of arithmetic. They are a special type of number because they cannot be broken down into smaller factors. 13 is prime because 13 is 1 times 13 (or 13 times 1), and that's it. There's no other way of expressing 13 as something times something. On the other hand, 12 is not prime because it splits into 2 times 6, or 3 times 4.

The first prime is 2. The next is 3. Then 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503 and so on. If you look at the first 10000 primes, you will see a list of numbers with no obvious pattern. There is even an air of mystery about them; if you didn't know they were prime numbers, you would probably have no idea how to continue the sequence. Indeed, if you do manage discover a simple pattern, you will have succeeded where some of the finest brains of all time have failed. For this is an area where mathematicians are well and truly baffled.

We do know fair amount about prime numbers, and an excellent starting point if you want to learn more about the subject is Chris Caldwell's web site: The Largest Known Primes. We know that the sequence of primes goes on for ever. We know that it thins out. The further you go, the rarer they get. We even have a simple formula for estimating roughly how many primes there are up to some large number without having to count them one by one. However, even though prime numbers have been the object of intense study by mathematicians for hundreds of years, there are still fairly basic questions which remain unanswered.

Prime Twins

If you look down the list of primes, you will quite often see two consecutive odd numbers, like 3 and 5, 5 and 7, 11 and 13, 17 and 19, or 29 and 31. We call these pairs of prime numbers {p, p + 2} prime twins.

The evidence suggests that, however far along the list of primes you care to look, you will always eventually find more examples of twins. Nevertheless, - and this may come as a surprise to you - it is not known whether this is in fact true. Possibly they come to an end. But it seems more likely that - like the primes - the sequence of prime twins goes on forever. However, Mathematics has yet to provide a rigorous proof.

One of the things mathematicians do when they don't understand something is produce bigger and better examples of the objects that are puzzling them. We run out of ideas, so we gather more data - and this is just what we are doing at this site; if you look ahead to section 2, you will see that I have collected together the ten largest known prime twins.

Prime Triplets

If you search the list for triples of primes {p, p + 2, p + 4}, you will not find very many. In fact there is only one, {3, 5, 7}, right at the beginning. And it's easy to see why. As G. H. Hardy & J. E. Littlewood observed [HL22], at least one of the three is divisible by 3.

Obviously it is asking too much to squeeze three primes into a range of four. However, if we increase the range to six and look for combinations {p, p + 2, p + 6} or {p, p + 4, p + 6}, we find plenty of examples, beginning with {5, 7, 11}, {7, 11, 13}, {11, 13, 17}, {13, 17, 19}, {17, 19, 23}, {37, 41, 43}, .... These are what we call prime triplets, and one of the main objectives of this site is to collect together all the largest known examples. Just as with twins, it is believed - but not known for sure - that the sequence of prime triplets goes on for ever.

Prime Quadruplets

Similar considerations apply to groups of four, where this time we require each of {p, p + 2, p + 6, p + 8} to be prime. Once again, it looks as if they go on indefinitely. The smallest is {5, 7, 11, 13}. We don't count {2, 3, 5, 7} even though it is a denser grouping. It is an isolated example which doesn't fit into the scheme of things. Nor, for more technical reasons, do we count {3, 5, 7, 11}.

The sequence continues with {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, .... The usual name is prime quadruplets, although I have also seen the terms full house, inter-decal prime quartet (!) and prime decade - a reference to the pattern made by their decimal digits. All primes greater than 5 end in one of 1, 3, 7 or 9, and the four primes in a (large) quadruplet always occur in the same ten-block. Hence there must be exactly one with each of these unit digits. And just to illustrate the point, here is another example; the smallest proven prime quadruplet of 2000 digits, found by Gerd Lamprecht in Oct 2017:

101999 + 205076414983951,
101999 + 205076414983953,
101999 + 205076414983957,
101999 + 205076414983959.

Prime k-tuplets

We can go on to define prime quintuplets, sextuplets, septuplets, octuplets, nonuplets, and so on. I had to go to the full Oxford English Dictionary for the last one - the Concise Oxford jumps from 'octuplets' to 'decuplets'. The OED also defines 'dodecuplets', but apparently there are no words for any of the others. Presumably I could make them up, but instead I shall use the number itself when I want to refer to, for example, prime 11-tuplets. I couldn't find the general term 'k-tuplets' in the OED either, but it is the word that seems to be in common use by the mathematical community.

For now, I will define a prime k-tuplet as a sequence of consecutive prime numbers such that the distance between the first and the last is in some sense as small as possible. If you think I am being too vague, there is a more precise definition later on.

At this site I have collected together what I believe to be the largest known prime k-tuplets for k = 2, 3, 4, ..., 20 and 21. I do not know of any prime k-tuplets for k greater than 21, except for the ones that occur near the beginning of the prime number sequence.

Notation

Multiplication is often denoted by an asterisk: xy is x times y.

For k > 2, the somewhat bizarre notation N + b1, b2, ..., bk is used (only in linked pages) to denote the k numbers {N + b1, N + b2, ..., N + bk}.

Prime twins are represented as N ± 1, which is short for N plus one and N minus one.

I also use the notation n# of Caldwell and Dubner [CD93] as a convenient shorthand for the product of all the primes less than or equal to n. Thus, for example, 20# = 2•3•5•7•11•13•17•19 = 9699690.

Finally ...

I would like to keep this site as up to date as possible. Therefore, can I urge you to please send any new, large prime k-tuplets to me. You can see what I mean by 'large' by studying the lists. If the numbers are not too big, say up to 500 digits, I am willing to double-check them myself. Otherwise I would appreciate some indication of how you proved that your numbers are true primes. Email address: pzktupel[at]pzktupel[dot]de.

2. The Largest Known Prime ~

Twins
Triplets
Quadruplets
Quintuplets
Sextuplets
Septuplets
Octuplets
Nonuplets
10-tuplets
11-tuplets
12-tuplets
13-tuplets
14-tuplets
15-tuplets
16-tuplets
17-tuplets
18-tuplets
19-tuplets
20-tuplets
21-tuplets

3. Summary

Largest known & early discovery of non-trivial prime k-tuplets to given pattern.

Early discovery with at least 100, 200, ..., 900 digits.
Early discovery with at least 1000, 2000, ..., 9000 digits.
Early discovery with at least 10000, 20000, ..., 90000 digits.
Early discovery with at least 100000 and more digits.

>
Smallest existing with 5 to 200 digits in step of 5.
Smallest existing with 100 to 2000 digits in step of 100.
Smallest existing with 1000 to 9000 digits in step of 1000.
Smallest existing with 10000 and 20000 digits.
Smallest existing googol prime k-tuplets.

Tables of values of π(x) up to π21(x)
Tables of values of πk(10n)  n=1..17, k=1..16

Initial members of prime k-tuplets
The smallest existing n-digit prime k-tuplets

First initial members of consecutive prime k-tuplets (PART I)
First initial members of consecutive prime k-tuplets (PART II)

Possible patterns & the Hardy-Littlewood constants of prime k-tuplets [HL22].


The Largest Known Twin Primes
Digits
When
Additions
2996863034895 • 21290000 ± 1
388342
19 Sep 2016
Tom Greer, TwinGen, PrimeGrid, LLR
3756801695685 • 2666669 ± 1
200700
26 Dec 2011
Timothy Winslow, TwinGen, PrimeGrid, LLR
65516468355 • 2333333 ± 1
100355
15 Aug 2009
Peter Kaiser, NewPGen, PrimeGrid, TPS, LLR
160204065 • 2262148 ± 1
78923
8 Jul 2021
Erwin Doescher, LLR
12770275971 • 2222225 ± 1
66907
4 Jul 2017
Bo Tornberg, TwinGen, LLR
12599682117 • 2211088 ± 1
63554
22 Feb 2022
Michael Kwok, PSieve, LLR
12566577633 • 2211088 ± 1
63554
22 Feb 2022
Michael Kwok, PSieve, LLR
70965694293 • 2200006 ± 1
60219
2 Apr 2016
S. Urushihata
66444866235 • 2200003 ± 1
60218
2 Apr 2016
S. Urushihata
4884940623 • 2198800 ± 1
59855
3 Jul 2015
Michael Kwok, PSieve, LLR
more

The Largest Known Prime Triplets
Digits
When
Additions
Certificates
4111286921397 • 266420 - 1 + d, d = 0, 2, 6
20008
24 Apr 2019
Peter Kaiser, Polysieve, LLR, Primo
click
6521953289619 • 255555 - 5 + d, d = 0, 4, 6
16737
30 Apr 2013
Peter Kaiser
click
56667641271 • 244441 - 1 + d, d = 0, 2, 6
13389
1 Apr 2022
Stephan Schöler, NewPGen, OpenPFGW; Oliver Kruse, Primo
click
4207993863 • 238624 - 1 + d, d = 0, 2, 6
11637
5 Jun 2021
Frank Doornink, NewPGen, LLR, Primo
click
14059969053 • 236672 - 5 + d, d = 0, 4, 6
11050
17 Jun 2018
Serge Batalov, NewPgen, OpenPFGW, Primo
click
3221449497221499 • 234567 - 1 + d, d = 0, 2, 6
10422
2 Sep 2015
Peter Kaiser, NewPGen, LLR, OpenPFGW, Primo
click
1288726869465789 • 234567 - 5 + d, d = 0, 4, 6
10421
23 Apr 2014
Peter Kaiser, Primo
click
647935598824239 • 233619 - 1 + d, d = 0, 2, 6
10136
22 May 2019
Peter Kaiser, Primo
click
209102639346537 • 233620 - 1 + d, d = 0, 2, 6
10135
22 May 2019
Peter Kaiser, Primo
click
185353103135997 • 233620 - 1 + d, d = 0, 2, 6
10135
22 May 2019
Peter Kaiser, Primo
click
more

The Largest Known Prime Quadruplets
Digits
When
Additions
Certificates
667674063382677 • 233608 - 1 + d, d = 0, 2, 6, 8
10132
27 Feb 2019
Peter Kaiser, Primo
click
4122429552750669 • 216567 - 1 + d, d = 0, 2, 6, 8
5003
10 Mar 2016
Peter Kaiser, GSIEVE, NewPGen, LLR, Primo
click
101406820312263 • 212042 - 1 + d, d = 0, 2, 6, 8
3640
13 Jun 2018
Serge Batalov, OpenPFGW, NewPGen, Primo
click
2673092556681 • 153048 - 4 + d, d = 0, 2, 6, 8
3598
14 Sep 2015
Serge Batalov, OpenPFGW, NewPGen, Primo
click
2339662057597 • 103490 + 1 + d, d = 0, 2, 6, 8
3503
21 Dec 2013
Serge Batalov, OpenPFGW, NewPGen, Primo
click
305136484659 • 211399 - 1 + d, d = 0, 2, 6, 8
3443
28 Sep 2013
Serge Batalov, OpenPFGW, NewPGen, Primo
click
722047383902589 • 211111 - 1 + d, d = 0, 2, 6, 8
3360
20 Apr 2013
Reto Keiser, NewPGen, OpenPFGW, Primo
click
585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8
3344
06 Mar 2022
Peter Kaiser, OpenPFGW, Primo
click
43697976428649 • 29999 - 1 + d, d = 0, 2, 6, 8
3024
26 Mar 2012
Peter Kaiser
click
102999 + 339930644528851 + d, d = 0, 2, 6, 8
3000
04 May 2022
Norman Luhn, NewPGen, OpenPFGW, Primo
click
more

The Largest Known Prime Quintuplets
Digits
When
Additions
Certificates
585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8, 12
3344
06 Mar 2022
Peter Kaiser, OpenPFGW, Primo
click
566761969187 • 4733# / 2 - 8 + d, d = 0, 4, 6, 10, 12
2034
06 Dec 2020
Serge Batalov, NewPGen, OpenPFGW, Primo
click
126831252923413 • 4657# / 273 + 1 + d, d = 0, 2, 6, 8, 12
2002
8 Nov 2020
Peter Kaiser, Primo
click
394254311495 • 3733# / 2 - 8 + d, d = 0, 4, 6, 10, 12
1606
Nov 2017
Serge Batalov, NewPGen, OpenPFGW, Primo
click
2316765173284 • 3600# + 16061 + d, d = 0, 2, 6, 8, 12
1543
16 Oct 2016
Norman Luhn, Primo
click
163252711105 • 3371# / 2 - 8 + d, d = 0, 4, 6, 10, 12
1443
1 Jan 2014
Serge Batalov, OpenPFGW, NewPGen, Primo
click
9039840848561 • 3299# / 35 - 5 + d, d = 0, 4, 6, 10, 12
1401
Dec 2013
Serge Batalov, OpenPFGW, NewPGen, Primo
click
699549860111847 • 24244 - 1 + d, d = 0, 2, 6, 8, 12
1293
3 Dec 2013
Reto Keiser, R. Gerbicz, OpenPFGW, Primo
click
405095429109490796 • 2683# + 16057 + d, d = 0, 4, 6, 10, 12
1150
4 Jul 2020
Michael Bell, Rieminer, ECPP-DJ
click
566650659276 • 2621# + 1615841 + d, d = 0, 2, 6, 8, 12
1117
Dec 2013
David Broadhurst, Primo, OpenPFGW
click
more

The Largest Known Prime Sextuplets
Digits
When
Additions
Certificates
23700 + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16
1114
8 Nov 2021
Vidar Nakling, Primo, Sixfinder
( based on Riecoin miners )
click
28993093368077 • 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16
1037
14 Mar 2016
Norman Luhn, APSIEVE, Primo
click
6646873760397777881866826327962099685830865900246688640856 • 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16
780
8 Nov 2018
Vidar Nakling, Primo
-
29720510172503062360713760607985203309940766118866743491802189150471978534404249 • 22299 +
14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16
772
28 Jan 2018
Riecoin #822096
-
29749903422302373222996698880833194129159047179535887991184960156219652236318921 • 22293 +
679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16
770
28 Sep 2017
Riecoin #793872
-
29696802688480280387313212926526693549449146292085717645262228449092881114972806 • 22290 +
1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16
769
25 Feb 2018
Riecoin #838224
-
29744205023784420961031622414734790416939049568996819659808238403983863222665068 • 22288 +
14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16
769
18 Feb 2018
Riecoin #834192
-
29707412718946949415029080194980493978605678414396606766712262274235284928962561 • 22278 +
21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16
766
14 Jan 2018
Riecoin #814032
-
29696978890366869883141509418765838581871522982358338407613039711378021084519043 • 22259 +
24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16
766
31 Dec 2017
Riecoin #805968
-
29691575669072177222494655186416928710256802541243921484227880404600991044790342 • 22259 +
22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16
760
16 Dec 2017
Riecoin #797904
-
more

The Largest Known Prime Septuplets
Digits
When
Additions
Certificates
113225039190926127209 • 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20
1002
27 Jan 2021
Peter Kaiser
click
3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473
9224962020153653925994202321897236902676229040360901005487309186655777663859063397693729163631275766
0779987530903845763711693853827939526026506444774774261236889041020217108597484837589978261046949778
7199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089
• 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20
527
16 Jun 2019
Vidar Nakling,
Rieminer 0.9, Primo
-
115828580393941 • 1200# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20
515
18 Jan 2018
Norman Luhn, Primo
-
4733578067069 • 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20
402
May 2016
Norman Luhn, Primo
-
697723422149271424870176724491962624555 • 701# + 145888993435301 + d, d = 0, 2, 6, 8, 12, 18, 20
332
28 Apr 2022
Michalis Christou, Rieminer
-
687001431518312990252195799540952 • 719# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20
331
25 Sep 2020
Michalis Christou, Rieminer
-
686636073174158279347746711902518 • 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20
331
25 Sep 2020
Michalis Christou, Rieminer
-
686488342697495738978150794512038 • 719# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20
331
25 Sep 2020
Michalis Christou, Rieminer
-
686305940768787196771563962517233 • 719# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20
331
25 Sep 2020
Michalis Christou, Rieminer
-
686129792256610907998640667932122 • 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20
331
25 Sep 2020
Michalis Christou, Rieminer
-
more

The Largest Known Prime Octuplets
Digits
When
Additions
697723422149271424870176724491962624555 • 701# + 145888993435301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
332
28 Apr 2022
Michalis Christou, Rieminer 0.91
6879356578124627875380298699944709053335 • 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
324
12 Mar 2021
Michalis Christou, Rieminer 0.91
54598824190010361875282469578684418459657573362461324471660422883073099662240278837985413217294784653805
• 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26
316
30 Oct 2021
Riecoin #1607166
237290937625019988409934680338216405908629349352492341129431599973490073614754863588338476036934867547671407908
• 487# + 1146773 + d, d = 0, 6, 8, 14, 18, 20, 24, 26
312
20 Oct 2021
Riecoin #1600958
188273324392097141944873869557423547058811920840483304365112457383885407879644413861445197917160744
• 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26
310
20 Oct 2021
Riecoin #1600978
6068138408292784654794269848877333341123929067736255007020032491702134706361073607222476583743922495929518535
• 487# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
310
20 Oct 2021
Riecoin #1600993
7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341
• 503# + 220469307413891 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
309
16 Feb 2022
Bielawski Mathematicians
6492845263546348546 • 700# + 226449521 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
309
19 Oct 2019
Thomas Nguyen, Rieminer 0.91
1047943643885526326864429446302233668685542252803581956414360456716437139890889113460138603012063154
• 503# + 145844141558231 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
308
16 Feb 2022
Riecoin #1669836
57365485417354499617885512657422186835780211950569618018314814739215714609244975671241124394854926629178629296
• 479# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
308
16 Feb 2022
Riecoin #1669754
more

The Largest Known Prime Nonuplets
Digits
When
Additions
7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341
• 503# + 220469307413891 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
309
16 Feb 2022
Bielawski Mathematicians
3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501
• 463# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
302
22 Aug 2021
Riecoin #1567399
1620259924615470570706663156278905026372754732844252658390408090245313172792664271166384219300680488342402961778
• 450# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
296
20 Aug 2021
Riecoin #1566093
387833514641724600357029749119397331285062620621983133723181869572568059514167753188325960698719230
• 467# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30
294
27 Oct 2021
Riecoin #1605403
352360483181346865458241271679723612698522403422825810225007396594617602134749714450854301781722077875166
• 457# + 302000014586509 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30
292
27 Dec 2021
Riecoin #1640522
40893595297845006551741048717748959451570266851095389722761855002653709793065456232477944049520841797242
• 457# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
291
7 Sep 2021
Riecoin #1576463
1285897436414229397879253137835709328420499854296690250107786136168118236699867925624486888053153988984555053
• 443# + 27899359258009 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30
291
19 Dec 2021
Riecoin #1636021
425637736526956247129170414648410773829792758837126862506126866133457379262600484880404875012755358905043168102
• 439# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
291
14 Mar 2022
Riecoin #1685127
1574649584907747555706133630785765300471148459969743147116987603964781873723920925748330533139472006902
• 457# + 145799289681161 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
290
6 Dec 2021
Riecoin #1628519
7524846006965529269346221744667651129320012118355693219288752951053384002012659523084964251323
• 467# + 145799289681161 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
290
20 Mar 2022
Riecoin #1688667
more

The Largest Known Prime 10-tuplets
Digits
When
Additions
14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550
• 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32
282
12 Sep 2021
Riecoin #1579367
290901656335108169864195656135043662615199446375386143995339722400236057821426952579658098504166333411889
• 401# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
269
27 Jul 2021
Riecoin #1551825
33521646378383216495527 • 331# + 4700094892301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
156
4 Apr 2020
Thomas Nguyen,
Rieminer 0.91, MPZ-APRCL
772556746441918 • 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32
136
9 Feb 2017
Norman Luhn
7425 • 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
120
27 May 2016
Roger Thompson
118557188915212 • 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
118
27 Jun 2014
Norman Luhn
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 53586844409797545 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
108
23 Sep 2019
Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 51143234991402697 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
108
23 Sep 2019
Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 50679161987995696 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
108
23 Sep 2019
Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 49561325184911775 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32
108
23 Sep 2019
Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo
more

The Largest Known Prime 11-tuplets
Digits
When
Additions
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 49376500222690335 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 46622982649030457 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 30796489110940369 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 27407861785763183 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 20731977215353082 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 20118509988610513 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 15866045335517629 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 5238271627884665 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
107
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 4471872451082759 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
107
28 May 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 1296173254392493 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
107
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
more

The Largest Known Prime 12-tuplets
Digits
When
Additions
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
+ 27407861785763183 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
108
23 Sep 2019
Peter Kaiser, David Stevens,
Polysieve, OpenPFGW, Primo
613176722801194 • 151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42
75
30 Sep 2014
Michael Stocker, Primo
467756 • 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
66
20 May 2014
Roger Thompson
9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
66
1 Oct 2021
Roger Thompson
9985397181 • 139# + 249386599747880711 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
66
1 Oct 2021
Roger Thompson
59125383480754 • 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42
61
9 Sep 2013
Michael Stocker
78989413043158 • 109# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
59
18 Jan 2010
Norman Luhn
450725899 • 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
56
4 Nov 2014
Martin Raab
14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
50
5 Feb 2013
Roger Thompson
8486221 • 107# + 4549290807806861 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
50
3 Jun 2006
Dirk Augustin & Jens Kruse Andersen
more

The Largest Known Prime 13-tuplets
Digits
When
Additions
9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48
66
1 Oct 2021
Roger Thompson
4135997219394611 • 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48
61
23 Mar 2017
Norman Luhn
14815550 • 107# + 4385574275277313 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48
50
5 Feb 2013
Roger Thompson
14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48
50
5 Feb 2013
Roger Thompson
61571 • 107# + 4803194122972361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48
48
7 Aug 2009
Jens Kruse Andersen
381955327397348 • 80# + 18393211 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48
46
28 Dec 2007
Norman Luhn
381955327397348 • 80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48
46
28 Dec 2007
Norman Luhn
100000000000000000000006149198224095343810309 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48
45
02 Mar 2022
Norman Luhn
100000000000000000000004356680452416578030761 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48
45
05 Feb 2022
Norman Luhn
100000000000000000000002004740564798426955633 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48
45
06 Mar 2022
Norman Luhn
more

The Largest Known Prime 14-tuplets
Digits
When
Additions
14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
50
5 Feb 2013
Roger Thompson
381955327397348 • 80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50
46
28 Dec 2007
Norman Luhn
1000000000000000014210159036148101380471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
40
10 Mar 2021
Norman Luhn
1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50
40
10 Mar 2021
Norman Luhn
10000000000009283441665311798539399 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50
35
18 Feb 2021
Norman Luhn
10000000000001275924044876917671361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
35
18 Feb 2021
Norman Luhn
26093748 • 67# + 383123187762431 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
33
8 Feb 2005
Christ van Willegen & Jens Kruse Andersen
108804167016152508211944400342691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
33
14 Apr 2008
Jens Kruse Andersen
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
33
14 Apr 2008
Jens Kruse Andersen
101885197790002105359911556070661 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
33
14 Apr 2008
Jens Kruse Andersen
more

The Largest Known Prime 15-tuplets
Digits
When
Additions
33554294028531569 • 61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56
40
25 Jan 2017
Norman Luhn
322255 • 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56
35
18 Nov 2016
Roger Thompson
10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56
35
4 Sep 2012
Roger Thompson
94 • 79# + 1341680294611244014367 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56
33
5 Feb 2021
Roger Thompson
3684 • 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56
33
5 Feb 2021
Roger Thompson
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56
33
14 Apr 2008
Jens Kruse Andersen
99999999948164978600250563546411 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56
32
29 Nov 2004
Jörg Waldvogel and Peter Leikauf
1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56
31
16 Oct 2003
Hans Rosenthal & Jens Kruse Andersen
1100916249233879857334075234831 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56
31
16 Oct 2003
Hans Rosenthal & Jens Kruse Andersen
1003234871202624616703163933857 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56
31
9 Aug 2012
Roger Thompson
more

The Largest Known Prime 16-tuplets
Digits
When
Additions
322255 • 73# + 1354238543317302647 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60
35
18 Nov 2016
Roger Thompson
94 • 79# + 1341680294611244014363 + d, d= 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60
33
5 Feb 2021
Roger Thompson
3684 • 73# + 880858118723497737827 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60
33
5 Feb 2021
Roger Thompson
1003234871202624616703163933853 + d, d= 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60
31
9 Aug 2012
Roger Thompson
11413975438568556104209245223 + d, d= 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60
29
2 Jan 2012
Roger Thompson
5867208169546174917450988007 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60
28
11 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
5621078036155517013724659017 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60
28
4 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
4668263977931056970475231227 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
4652363394518920290108071177 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
4483200447126419500533043997 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
more

The Largest Known Prime 17-tuplets
Digits
When
Additions
3684 • 73# + 880858118723497737821 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66
33
5 Feb 2021
Roger Thompson
100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66
30
19 Feb 2013
Roger Thompson
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66
29
2 Jan 2012
Roger Thompson
11410793439953412180643704677 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66
29
2 Jan 2012
Roger Thompson
5867208169546174917450988001 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66
28
11 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66
28
11 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
5621078036155517013724659011 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66
28
4 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66
28
4 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
4668263977931056970475231221 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
more

The Largest Known Prime 18-tuplets
Digits
When
Additions
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
11 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
4 Mar 2014
Raanan Chermoni & Jaroslaw Wroblewski
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
4652363394518920290108071167 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
4483200447126419500533043987 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
4 Jan 2014
Raanan Chermoni & Jaroslaw Wroblewski
3361885098594416802447362317 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
30 Jul 2013
Raanan Chermoni & Jaroslaw Wroblewski
3261917553005305074228431077 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
30 Jul 2013
Raanan Chermoni & Jaroslaw Wroblewski
3176488693054534709318830357 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
30 Jul 2013
Raanan Chermoni & Jaroslaw Wroblewski
2650778861583720495199114537 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
25 Feb 2013
Raanan Chermoni & Jaroslaw Wroblewski
2406179998282157386567481197 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70
28
31 Dec 2012
Raanan Chermoni & Jaroslaw Wroblewski
more

The Largest Known Prime 19-tuplets
Digits
When
Additions
622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76
30
27 Dec 2018
Raanan Chermoni & Jaroslaw Wroblewski
248283957683772055928836513597 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76
30
1 Aug 2016
Raanan Chermoni & Jaroslaw Wroblewski
138433730977092118055599751677 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76
30
8 Oct 2015
Raanan Chermoni & Jaroslaw Wroblewski
39433867730216371575457664407 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76
29
8 Jan 2015
Raanan Chermoni & Jaroslaw Wroblewski
2406179998282157386567481191 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76
28
31 Dec 2012
Raanan Chermoni & Jaroslaw Wroblewski
2348190884512663974906615481 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76
28
17 Dec 2012
Raanan Chermoni & Jaroslaw Wroblewski
917810189564189435979968491 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76
27
29 May 2011
Raanan Chermoni & Jaroslaw Wroblewski
656632460108426841186109951 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76
27
19 Feb 2011
Raanan Chermoni & Jaroslaw Wroblewski
630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76
27
9 Feb 2011
Raanan Chermoni & Jaroslaw Wroblewski
{37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
more

The Largest Known Prime 20-tuplets
Digits
When
Additions
1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80
31
23 May 2021
Raanan Chermoni & Jaroslaw Wroblewski
1188350591359110800209379560799 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80
31
21 Jan 2021
Raanan Chermoni & Jaroslaw Wroblewski
1153897621507935436463788957529 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80
31
26 Dec 2020
Raanan Chermoni & Jaroslaw Wroblewski
1135540756371356698957890225821 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80
31
19 Dec 2020
Raanan Chermoni & Jaroslaw Wroblewski
1126002593922465663847897293731 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80
31
17 Nov 2020
Raanan Chermoni & Jaroslaw Wroblewski
1094372814043722195189448411199 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80
31
20 Oct 2020
Raanan Chermoni & Jaroslaw Wroblewski
1060475118776959297139870952701 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80
31
18 Sep 2020
Raanan Chermoni & Jaroslaw Wroblewski
999627565307688186459783232931 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80
30
19 Jun 2020
Raanan Chermoni & Jaroslaw Wroblewski
957278727962618711849051282459 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80
30
23 Mar 2020
Raanan Chermoni & Jaroslaw Wroblewski
839013472011818416634745523991 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80
30
28 Oct 2020
Raanan Chermoni & Jaroslaw Wroblewski
more

The Largest Known Prime 21-tuplets
Digits
When
Additions
622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84
30
27 Dec 2018
Raanan Chermoni & Jaroslaw Wroblewski
248283957683772055928836513589 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84
30
1 Aug 2016
Raanan Chermoni & Jaroslaw Wroblewski
138433730977092118055599751669 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84
31
8 Oct 2015
Raanan Chermoni & Jaroslaw Wroblewski
39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84
29
8 Jan 2015
Raanan Chermoni & Jaroslaw Wroblewski
{29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
more


4. Mathematical Background

Definition

A prime k-tuplet is a sequence of k consecutive prime numbers such that in some sense the difference between the first and the last is as small as possible. The idea is to generalise the concept of prime twins.

More precisely: We first define s(k) to be the smallest number s for which there exist k integers b1 < b2 < ... < bk, bkb1 = s and, for every prime q, not all the residues modulo q are represented by b1, b2, ..., bk. A prime k-tuplet is then defined as a sequence of consecutive primes {p1, p2, ..., pk} such that for every prime q, not all the residues modulo q are represented by p1, p2, ..., pk, pkp1 = s(k). Observe that the definition might exclude a finite number (for each k) of dense clusters at the beginning of the prime number sequence - for example, {97, 101, 103, 107, 109} satisfies the conditions of the definition of a prime 5-tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.

Patterns of Prime k-tuplets

The simplest case is s(2) = 2, corresponding to prime twins: {p, p + 2}. Next, s(3) = 6 and two types of prime triplets: {p, p + 2, p + 6} and {p, p + 4, p + 6}, followed by s(4) = 8 with just one pattern: {p, p + 2, p + 6, p + 8} of prime quadruplets. The sequence continues with s(5) = 12, s(6) = 16, s(7) = 20, s(8) = 26, s(9) = 30, s(10) = 32, s(11) = 36, s(12) = 42, s(13) = 48, s(14) = 50, s(15) = 56, s(16) = 60, s(17) = 66 and so on. It is number A008407 in N.J.A. Sloane's On-line Encyclopedia of Integer Sequences.

Primality Proving

In keeping with similar published lists, I have decided not to accept anything other than true, proven primes. Numbers which have merely passed the Fermat test, aN = a (mod N), will need to be validated. If N − 1 or N + 1 is sufficiently factorized (usually just under a third), the methods of Brillhart, Lehmer and Selfridge [BLS75] will suffice. Otherwise the numbers may have to be subjected to a general primality test, such as the Jacobi sum test of Adleman, Pomerance, Rumely, Cohen and Lenstra (APRT-CLE in UBASIC, for example), or one of the elliptic curve primality proving programs: Atkin and Morain's ECPP, or its successor, Franke, Kleinjung, Wirth and Morain's FAST-ECPP, or Marcel Martin's Primo.

Primes

Euclid proved that there are infinitely many primes. Paulo Ribenboim [Rib95] has collected together a considerable number of different proofs of this important theorem. My favourite (which is not in Ribenboim's book) goes like this: We have

p prime 1/(1 − 1/p2) = ∑n = 1 to ∞ 1/n2 = π2/6.

But π2 is irrational; so the product on the left cannot have a finite number of factors.

In its simplest form, the prime number theorem states that the number of primes less than x is asymptotic to x/(log x). This was first proved by Hadamard and independently by de la Vallee Poussin in 1896. Later, de la Vallee Poussin found a better estimate:

u = 0 to x du/(log u) + error term,

where the error term is bounded above by A x exp(−B √(log x)) for some constants A and B. With more work (H.-E. Richert, 1967), √(log x) in this last expression can be replaced by (log x)3/5(log log x)−1/5. The most important unsolved conjecture of prime number theory, indeed, all of mathematics - the Riemann Hypothesis - asserts that the error term can be bounded by a function of the form Ax log x.

The Twin Prime Conjecture

G.H. Hardy & J.E. Littlewood did the first serious work on the distribution of prime twins. In their paper 'Some problems of Partitio Numerorum: III...' [HL22], they conjectured a formula for the number of twins between 1 and x:

2 C2 x / (log x)2,

where C2 = ∏p prime, p > 2 p(p − 2) / (p − 1)2 = 0.66016... is known as the twin prime constant.

V. Brun showed that the sequence of twins is thin enough for the series ∑p and p + 2 prime 1 / p to converge. The twin prime conjecture states that the sum has infinitely many terms. The nearest to proving the conjecture is Jing-Run Chen's result that there are infinitely many primes p such that p + 2 is either prime or the product of two primes [HR73].

The Hardy-Littlewood Prime k-tuple Conjecture

The Partitio Numerorum: III paper [HL22] goes on to formulate a general conjecture concerning the distribution of arbitrary groups of prime numbers (The k-tuplets of this site are special cases): Let b1, b2, ..., bk be k distinct integers. Then the number of groups of primes N + b1, N + b2, ..., N + bk between 2 and x is approximately

Hk Cku = 2 to x du / (log u)k,

where

Hk = ∏p prime, pk pk − 1 (pv) / (p − 1)kp prime, p > k, p|D (pv) / (pk),

Ck = ∏p prime, p > k pk − 1 (pk) / (p − 1)k,

v = v(p) is the number of distinct remainders of b1, b2, ..., bk modulo p and D is the product of the differences |bi − bj|, 1 ≤ i < j ≤ k.

The first product in Hk is over the primes not greater than k, the second is over the primes greater than k which divide D and the product Ck is over all primes greater than k. If you put k = 2, b1 = 0 and b2 = 2, then v(2) = 1, v(p) = p − 1 for p > 2, H2 = 2, and Ck = C2, the twin prime constant given above.

It is worth pointing out that with modern mathematical software the prime k-tuplet constants Ck can be determined to great accuracy. The way not to do it is to use the defining formula. Unless you are very patient, calculating the product over a sufficient number of primes for, say, 20 decimal place accuracy would not be feasible. Instead there is a useful transformation originating from the product formula for the Riemann ζ function:

log Ck = − ∑n = 2 to ∞ log [ζ(n) ∏p prime, pk (1 − 1/pn)] / nd|n μ(n/d) (kdk).

5. References

[BLS75] John Brillhart, D.H. Lehmer & J.L. Selfridge, New primality criteria and factorizations of 2m ± 1, Math. Comp. 29 (1975), 620-647.

[CD93] C.K.Caldwell & H. Dubner, Primorial, factorial and multifactorial primes, Math. Spectrum 26 (1993/94), 1-7.

[F97f] Tony Forbes, Prime 17-tuplet, NMBRTHRY Mailing List, September 1997.

[F02] Tony Forbes, Titanic prime quintuplets, M500 189 (December, 2002), 12-13.

[F09] Tony Forbes, Gigantic prime triplets, M500 226 (February, 2009), 18-19.

[Guy94] Richard K. Guy, Unsolved Problems in Number Theory, second edn., Springer-Verlag, New York 1994.

[HL22] G. H. Hardy and J. E. Littlewood, Some problems of Partitio Numerorum: III; on the expression of a number as a sum of primes, Acta Mathematica 44 (1922), 1-70.

[HR73] H. Halberstam and H.-E Richert, Sieve Methods, Academic Press, London 1973.

[Rib95] P. Ribenboim, The New Book of Prime Number Records, 3rd edn., Springer-Verlag, New York 1995

[R96a] Warut Roonguthai, Prime quadruplets, M500 148 (February 1996), 9.

[R96b] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1996.

[R96c] Warut Roonguthai, Large prime quadruplets, M500, 153 (December, 1996), 4-5.

[R97a] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1997.

[R97b] Warut Roonguthai, Large prime quadruplets, M500 158 (November 1997), 15.