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Prime k-tuplets

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@pzktupel.de

This site address: https://pzktupel.de/ktuplets.htm
Also I maintained the pages from Jens Kruse Andersen and Dirk Augustin: CPAP, AP, Cunningham Chain Records & Simultaneous Primes click

Additions ( last updated: 4 December 2021 )

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

Contents

  1. Introduction
  2. The Largest Known Prime Twins
  3. The Largest Known Prime Triplets
  4. The Largest Known Prime Quadruplets
  5. The Largest Known Prime Quintuplets
  6. The Largest Known Prime Sextuplets
  7. The Largest Known Prime Septuplets
  8. The Largest Known Prime Octuplets
  9. The Largest Known Prime 9-tuplets
  10. The Largest Known Prime 10-tuplets
  11. The Largest Known Prime 11-tuplets
  12. The Largest Known Prime 12-tuplets
  13. The Largest Known Prime 13-tuplets
  14. The Largest Known Prime 14-tuplets
  15. The Largest Known Prime 15-tuplets
  16. The Largest Known Prime 16-tuplets
  17. The Largest Known Prime 17-tuplets
  18. The Largest Known Prime 18-tuplets
  19. The Largest Known Prime 19-tuplets
  20. The Largest Known Prime 20-tuplets
  21. The Largest Known Prime 21-tuplets
  22. Summary
  23. Mathematical Background
  24. References
  25. List of all possible patterns of prime k-tuplets & the Hardy-Littlewood constants pertaining to the of prime k-tuplets [HL22]
  26. Tables of values of π(x) up to π21(x)
  27. Tables of values of πk(10n)  n=1..16
  28. The big database of "The smallest prime k-tuplets"
  29. The big database of "The smallest n-digit prime k-tuplets"
  30. Collection of Prime Page Links
  31. 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 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: x*y 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@pzktupel.de.

    2. 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
    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
    2003663613 * 2195000 ± 1
    58711
    15 Jan 2007
    Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon,
    Michael Kwok, Andrea Pacini & Rytis Slatkevicius
    17976255129 * 2183241 ± 1
    55172
    10 May 2021
    Frank Doornink, TwinGen, OpenPFGW
    more

    3. The Largest Known Prime Triplets
    Digits
    When
    Additions
    4111286921397 * 266420 - 1 + d, d = 0, 2, 6
    20008
    24 Apr 2019
    Peter Kaiser, Polysieve, LLR, Primo
    6521953289619 * 255555 - 5 + d, d = 0, 4, 6
    16737
    30 Apr 2013
    Peter Kaiser
    4207993863 * 238624 - 1 + d, d = 0, 2, 6
    11637
    5 Jun 2021
    Frank Doornink, NewPGen, LLR, Primo
    14059969053 * 236672 - 5 + d, d = 0, 4, 6
    11050
    17 Jun 2018
    Serge Batalov, NewPgen, OpenPFGW, Primo
    3221449497221499 * 234567 - 1 + d, d = 0, 2, 6
    10422
    2 Sep 2015
    Peter Kaiser, NewPGen, LLR, OpenPFGW
    1288726869465789 * 234567 - 5 + d, d = 0, 4, 6
    10421
    23 Apr 2014
    Peter Kaiser
    647935598824239 * 233619 - 1 + d, d = 0, 2, 6
    10136
    22 May 2019
    Peter Kaiser, Primo
    209102639346537 * 233620 - 1 + d, d = 0, 2, 6
    10135
    22 May 2019
    Peter Kaiser, Primo
    185353103135997 * 233620 - 1 + d, d = 0, 2, 6
    10135
    22 May 2019
    Peter Kaiser, Primo
    162615027598677 * 233620 - 1 + d, d = 0, 2, 6
    10135
    22 May 2019
    Peter Kaiser, Primo
    more

    4. The Largest Known Prime Quadruplets
    Digits
    When
    Additions
    667674063382677 * 233608 - 1 + d, d = 0, 2, 6, 8
    10132
    27 Feb 2019
    Peter Kaiser, Primo
    4122429552750669 * 216567 - 1 + d, d = 0, 2, 6, 8
    5003
    10 Mar 2016
    Peter Kaiser, GSIEVE, NewPGen, LLR, Primo
    101406820312263 * 212042 - 1 + d, d = 0, 2, 6, 8
    3640
    13 Jun 2018
    Serge Batalov, OpenPFGW, NewPGen, Primo
    2673092556681 * 153048 - 4 + d, d = 0, 2, 6, 8
    3598
    14 Sep 2015
    Serge Batalov, OpenPFGW, NewPGen, Primo
    2339662057597 * 103490 + 1 + d, d = 0, 2, 6, 8
    3503
    21 Dec 2013
    Serge Batalov, OpenPFGW, NewPGen, Primo
    305136484659 * 211399 - 1 + d, d = 0, 2, 6, 8
    3443
    28 Sep 2013
    Serge Batalov, OpenPFGW, NewPGen, Primo
    722047383902589 * 211111 - 1 + d, d = 0, 2, 6, 8
    3360
    20 Apr 2013
    Reto Keiser, NewPGen, PFGW, Primo
    43697976428649 * 29999 - 1 + d, d = 0, 2, 6, 8
    3024
    26 Mar 2012
    Peter Kaiser
    46359065729523 * 28258 - 1 + d, d = 0, 2, 6, 8
    2500
    20 Nov 2011
    Reto Keiser, NewPGen, PFGW, Primo
    1367848532291 * 5591# / 35 - 1 + d, d = 0, 2, 6, 8
    2401
    21 Aug 2011
    Norman Luhn, NewPGen, PFGW, Primo
    more

    5. The Largest Known Prime Quintuplets
    Digits
    When
    Additions
    566761969187 * 4733# / 2 - 8 + d, d = 0, 4, 6, 10, 12
    2034
    6 Dec 2020
    Serge Batalov, NewPGen, OpenPFGW, Primo
    126831252923413 * 4657# / 273 + 1 + d, d = 0, 2, 6, 8, 12
    2002
    8 Nov 2020
    Peter Kaiser, Primo
    394254311495 * 3733# / 2 - 8 + d, d = 0, 4, 6, 10, 12
    1606
    Nov 2017
    Serge Batalov, NewPGen, OpenPFGW, Primo
    2316765173284 * 3600# + 16061 + d, d = 0, 2, 6, 8, 12
    1543
    16 Oct 2016
    Norman Luhn, Primo
    163252711105 * 3371# / 2 - 8 + d, d = 0, 4, 6, 10, 12
    1443
    1 Jan 2014
    Serge Batalov, OpenPFGW, NewPGen, Primo
    9039840848561 * 3299# / 35 - 5 + d, d = 0, 4, 6, 10, 12
    1401
    Dec 2013
    Serge Batalov, OpenPFGW, NewPGen, Primo
    699549860111847 * 24244 - 1 + d, d = 0, 2, 6, 8, 12
    1293
    3 Dec 2013
    Reto Keiser, R. Gerbicz, PFGW, Primo
    405095429109490796 * 2683# + 16057 + d, d = 0, 4, 6, 10, 12
    1150
    4 Jul 2020
    Michael Bell, Rieminer, ECPP-DJ
    566650659276 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12
    1117
    Dec 2013
    David Broadhurst, Primo, OpenPFGW
    554729409262 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12
    1117
    Dec 2013
    David Broadhurst, Primo, OpenPFGW
    more

    6. The Largest Known Prime Sextuplets
    Digits
    When
    Additions
    23700 + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16
    1114
    8 Nov 2021
    Vidar Nakling, Primo, Sixfinder
    ( based on Riecoin miners )
    28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16
    1037
    14 Mar 2016
    Norman Luhn, APSIEVE, Primo
    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

    7. The Largest Known Prime Septuplets
    Digits
    When
    Additions
    113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20
    1002
    27 Jan 2021
    Peter Kaiser
    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
    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
    685817639451814894948541841801955 * 719# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20
    331
    25 Sep 2020
    Michalis Christou, Rieminer
    more

    8. The Largest Known Prime Octuplets
    Digits
    When
    Additions
    6879356578124627875380298699944709053335 * 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
    324
    12 Mar 2021
    Michalis Christou
    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
    6492845263546348546 * 700# + 226449521 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
    309
    19 Oct 2019
    Thomas Nguyen, Rieminer 0.91
    652860139668148506027538015230153318100639856923430236186190699612250481487112024837265410828490443427344306052
    * 467# + 1418575498583 + d, d = 0, 6, 8, 14, 18, 20, 24, 26
    307
    26 Oct 2021
    Riecoin #1604890
    1003196298617375646416642308462802536422799049663270195408652126607761421584998613928566265881964993799
    * 491# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26
    306
    26 Oct 2021
    Riecoin #1604885
    6906530913807107618746553985178874306453389164948020895483633515626362605252139055492044931282748833678120293
    * 467# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
    305
    22 Aug 2021
    Riecoin #1567265
    359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26
    304
    4 Jul 2017
    Norman Luhn, VFYPR
    more

    9. The Largest Known Prime Nonuplets
    Digits
    When
    Additions
    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
    40893595297845006551741048717748959451570266851095389722761855002653709793065456232477944049520841797242
    * 457# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
    291
    7 Sep 2021
    Riecoin #1576463
    732510298464897302406863094406522970022698347078480858977704780162529246459062655850703116903119380663871601
    * 440# + 114521428533971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
    288
    20 Aug 2021
    Riecoin #1558658
    115750297122237222935734922477288507624860627469651014827576218647836337473108302145884182747351935638522131
    * 439# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
    287
    24 Aug 2021
    Riecoin #1568122
    147807759781711968865389529272742298979147962123222700107620515659067459819265042300840259710140
    * 461# + 114189340938131 d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
    286
    26 Nov 2021
    Riecoin #1622250
    13941780746436414018698387636102779229293375330619030932204602199443540774826962448707456602381399834618
    * 443# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
    286
    20 Aug 2021
    Riecoin #1548122
    1519007803707207636172890273323957030631282142918279301959878723805156323295860440667348707667809976826323916
    * 433# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
    285
    20 Aug 2021
    Riecoin #1538130
    2013582968688100254021601080203506492283895313221178016484151445253584656191978234310137219527
    * 461# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30
    284
    24 Aug 2021
    Riecoin #1568391
    more

    10. The Largest Known Prime Decuplets
    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, PFGW, 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, PFGW, 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, PFGW, 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, PFGW, Primo
    more

    11. 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, PFGW, 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, PFGW, 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, PFGW, 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, PFGW, 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, PFGW, 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, PFGW, 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, PFGW, 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, PFGW, Primo
    13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
    + 4471872451082759 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
    107
    23 Sep 2019
    Peter Kaiser, David Stevens,
    Polysieve, PFGW, 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, PFGW, Primo
    more

    12. The Largest Known Prime Dodecuplets
    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, PFGW, 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

    13. 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
    1000000000000000027545153594708289884461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48
    40
    13 Jul 2021
    Norman Luhn
    1000000000000000014210159036148101380473 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48
    40
    13 Jul 2021
    Norman Luhn
    1000000000000000014210159036148101380471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48
    40
    13 Jul 2021
    Norman Luhn
    more

    14. 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
    Feb 2021
    Norman Luhn
    10000000000001275924044876917671361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50
    35
    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

    15. 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
    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
    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

    16. 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

    17. 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

    18. 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

    19. 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

    20. 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

    21. 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

    22. Summary

    The largest known prime k-tuplets
    k Digits Prime k-tuplet Who When
    1 24862048 282589933 − 1 P. Laroche, G. Woltman, S. Kurowski,
    A. Blosser,et al (GIMPS)
    21 Dec 2018
    2 388342 2996863034895 * 21290000 ± 1 Tom Greer, TwinGen,
    PrimeGrid, LLR
    19 Sep 2016
    3 20008 4111286921397 * 266420 - 1 + d, d = 0, 2, 6 Peter Kaiser, Polysieve, LLR, Primo 24 Apr 2019
    4 10132 667674063382677 * 233608 -1 + d, d = 0, 2, 6, 8 Peter Kaiser, Primo 27 Feb 2019
    5 2034 566761969187 * 4733#/2 - 8 + d, d = 0, 4, 6, 10, 12 Serge Batalov, NewPGen,
    OpenPFGW, Primo
    6 Dec 2020
    6 1114 23700 + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16 Vidar Nakling, Primo, Sixfinder 8 Nov 2021
    7 1002 113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 Peter Kaiser 27 Jan 2021
    8 324 6879356578124627875380298699944709053335 * 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 Michalis Christou, Rieminer 0.91 12 Mar 2021
    9 302 3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501
    * 463# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30
    Riecoin #1567399, PrimaPoolSolo 22 Aug 2021
    10 282 14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550
    * 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32
    Riecoin #1579367, PrimaPoolSolo 12 Sep 2021
    11 108 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
    + 49376500222690335 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36
    Peter Kaiser, David Stevens,
    Polysieve, OpenPFGW, Primo
    23 Sep 2019
    12 108 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
    + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
    Peter Kaiser, David Stevens,
    Polysieve, OpenPFGW, Primo
    23 Sep 2019
    13 66 9985637467 * 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 Roger Thompson 1 Oct 2021
    14 50 14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50Roger Thompson5 Feb 2013
    15 40 33554294028531569 * 61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56Norman Luhn25 Jan 2017
    16 35 322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 Roger Thompson 18 Nov 2016
    17 33 3684 * 73# + 880858118723497737821 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 Roger Thompson 5 Feb 2021
    18 28 5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 Raanan Chermoni &
    Jaroslaw Wroblewski
    Mar 2014
    19 30 622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76 Raanan Chermoni &
    Jaroslaw Wroblewski
    27 Dec 2018
    20 31 1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 Raanan Chermoni &
    Jaroslaw Wroblewski
    23 May 2021
    21 30 622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84 Raanan Chermoni &
    Jaroslaw Wroblewski
    27 Dec 2018
    22 2 {23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} - -
    23 2 {19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} - -
    24 - There are no known prime 24-tuplets - -

    Early discovery of a non-trivial prime k-tuplet
    k Digits Prime k-tuplet Who When
    <12 - No reliable information - -
    12 13 1418575498567 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 D. Betsis & S. Säfholm 1982
    13 14 10527733922579 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 D. Betsis & S. Säfholm 1982
    14 17 21817283854511261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 D. Betsis & S. Säfholm 1982
    15 21 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 Jörg Waldvogel 1996
    16 21 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 Jörg Waldvogel 1996
    17 22 1620784518619319025971 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 Jörg Waldvogel 1997
    18 25 2845372542509911868266807 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 Jörg Waldvogel & Peter Leikauf 14 Nov 2000
    19 27 630134041802574490482213901 +d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 Raanan Chermoni & Jaroslaw Wroblewski 9 Feb 2011
    20 28 3941119827895253385301920029 +d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 Raanan Chermoni & Jaroslaw Wroblewski 24 Jun 2014
    21 29 39433867730216371575457664399 +d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 Raanan Chermoni & Jaroslaw Wroblewski 8 Jan 2015

    Early discovery of 100 digits
    k Digits Prime k-tuplet Who When
    1 157 2521 − 1 R. M. Robinson 30 Jan 1952
    2-5 - No reliable information - -
    6 133 2 * 10132 + 75543532187 + d, d = 0, 4, 6, 10, 12, 16 Tony Forbes Apr 1994
    7 104 4293326603 * 233# + 399389 + d, d = 0, 2, 8, 12, 14, 18, 20 Radoslaw Naleczynski 31 Dec 1998
    8 110 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 Norman Luhn 23 Feb 2001
    9 110 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 Norman Luhn 23 Feb 2001
    10 103 72613488698235 * 227# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 Norman Luhn 11 Apr 2004
    11 104 24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 Norman Luhn & Jens Kruse Andersen 18 Aug 2004
    12 108 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501
    + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42
    Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 23 Sep 2019

    Early discovery of 1,000 digits
    k Digits Prime k-tuplet Who When
    1 1332 24423 − 1 Alexander Hurwitz 3 Nov 1961
    2 1040 256200945 * 23426 ± 1 Oliver Atkin & N. W. Rickert 1980
    3 1083 437850590 * (23567 − 21189) − 6 * 21189 - 5 + d, d = 0, 4, 6 Tony Forbes Dec 1996
    4 1004 76912895956636885 * (23279 − 21093) − 6 * 21093 - 7 + d, d = 0, 2, 6, 8 Tony Forbes 16 Sep 1998
    5 1034 31969211688 * 2400# + 16061 + d, d = 0, 2, 6, 8, 12 Norman Luhn 30 Jul 2002
    6 1037 28993093368077 * 2400# + 19417 + d, d = 0, 2, 6, 8, 12, 16 Norman Luhn, APSIEVE, Primo 14 Mar 2016
    7 1002 113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 Peter Kaiser 27 Jan 2021

    Early discovery of 10,000 digits
    k Digits Prime k-tuplet Who When
    1 13395 244497 − 1 Harry Nelson & David Slowinski 8 Apr 1979
    2 11713 242206083 * 238880 ± 1 H. K. Indlekofer & A. Járai Nov 1995
    3 10047 2072644824759 * 233333 - 1 + d, d = 0, 2, 6 Norman Luhn, François Morain, FastECPP 17 Nov 2008
    4 10132 667674063382677 * 233608 - 1 + d, d = 0, 2, 6, 8 Peter Kaiser, Primo 27 Feb 2019

    Early discovery of 100,000 digits
    k Digits Prime k-tuplet Who When
    1 227832 2756839 − 1 David Slowinski & Paul Gage 19 Feb 1992
    2 100355 65516468355 * 2333333 ± 1 Peter Kaiser, NewPGen, PRIMEGRID, TPS, LLR 15 Aug 2009

    Early discovery of 1,000,000 digits
    k Digits Prime k-tuplet Who When
    1 2098960 26972593 − 1 Nayan Hajratwala, George Woltman, Scott Kurowski et al (GIMPS) 1 Jun 1999

    Early discovery of 10,000,000 digits
    k Digits Prime k-tuplet Who When
    1 12978189 243112609 − 1 Edson Smith, George Woltman, Scott Kurowski et al (GIMPS) 23 Aug 2008

    23. 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).

    24. 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.

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