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

New site address: http://www.pzktupel.de/ktuplets.htm.

Additions ( last updated: 13 Sep 2021 )

Complete site history

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
  26. The Hardy-Littlewood constants pertaining to the distribution of prime k-tuplets [HL22]
  27. List of the smallest prime k-tuplets
  28. The "Smallest prime k-tuplets" database
  29. 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: anthony.d.forbes@gmail.com.

    2. The Largest Known Prime Twins

    2996863034895 * 21290000 ± 1 (388342 digits, Sep 2016, Tom Greer, TwinGen, PRIMEGRID, LLR)

    3756801695685 * 2666669 ± 1 (200700 digits, Dec 2011, Timothy Winslow, TwinGen, PRIMEGRID, LLR)

    65516468355 * 2333333 ± 1 (100355 digits, Aug 2009, Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR)

    160204065 * 2262148 ± 1 (78923 digits, Aug 2021, Erwin Doescher, LLR)

    12770275971 * 2222225 ± 1 (66907 digits, Jul 2017, Bo Tornberg, TwinGen, LLR TWIN)

    70965694293 * 2200006 ± 1 (60219 digits, Apr 2016, S. Urushihata)

    66444866235 * 2200003 ± 1 (60218 digits, Apr 2016, S. Urushihata)

    4884940623 * 2198800 ± 1 (59855 digits, Jul 2015, Kwok, PSIEVE, LLR)

    2003663613 * 2195000 ± 1 (58711 digits, Jan 2007, Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Michaek Kwok, Andrea Pacini, Rytis Slatkevicius)

    17976255129 * 2183241 ± 1 (55172 digits, May 2021, Frank Doornink, TwinGen, OpenPFGW)

    See Chris Caldwell, The Largest Known Primes for further (and possibly more up to date) information.

    3. The Largest Known Prime Triplets

    4111286921397 * 266420 + d, d = −1, 1, 5 (20008 digits, 24 Apr 2019, Peter Kaiser, POLYSIEVE, LLR, PRIMO)

    6521953289619 * 255555 + d, d = −5, −1, 1 (16737 digits, Apr 2013, Peter Kaiser)

    4207993863 * 238624 + d, d = −1, 1, 5 (11637 digits, Jun 2021, Frank Doornink, NEWGEN, LLR, PRIMO)

    3221449497221499 * 234567 + d, d = −1, 1, 5 (10422 digits, Sep 2015, Peter Kaiser, NEWGEN, LLR, OpenPFGW, )

    1288726869465789 * 234567 + d, d = −5, −1, +1 (10421 digits, Apr 2014, Peter Kaiser)

    647935598824239 * 233619 + d, d = −1, 1, 5 (10136 digits, 22 May 2019, Peter Kaiser, PRIMO)

    209102639346537 * 233620 + d, d = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

    185353103135997 * 233620 + d, d = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

    162615027598677 * 233620 + d, d = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

    667674063382677 * 233608 + d, d = 1, 5, 7 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)

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    4. The Largest Known Prime Quadruplets

    667674063382677 * 233608 + d, d = −1, 1, 5, 7 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)

    4122429552750669 * 216567 + d, d = −1, 1, 5, 7 (5003 digits, Mar 2016, Peter Kaiser, GSIEVE, NewPGen, LLR, PRIMO)

    101406820312263 * 212042 + d, d = -1, 1, 5, 7 (3640 digits, Jun 2018, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

    2673092556681 * 153048 + d, d = −4, −2, 2, 4 (3598 digits, Sep 2015, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

    2339662057597 * 103490 + d, d = 1, 3, 7, 9 (3503 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

    305136484659 * 211399 + d, d = −1, 1, 5, 7 (3443 digits, Sep 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

    722047383902589 * 211111 + d, d = −1, 1, 5, 7 (3360 digits, Apr 2013, Reto Keiser, NEWPGEN, PFGW, PRIMO)

    43697976428649 * 29999 + d, d = −1, 1, 5, 7 (3024 digits, Mar 2012, Peter Kaiser)

    46359065729523 * 28258 + d, d = −1, 1, 5, 7 (2500 digits, Nov 2011, Reto Keiser, NEWPGEN, PFGW, PRIMO)

    1367848532291 * 5591# / 35 + d, d = −1, 1, 5, 7 (2401 digits, Aug 2011, Norman Luhn, NEWPGEN, PFGW, PRIMO)

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    5. The Largest Known Prime Quintuplets

    566761969187 * 4733#/2 + d, d = −8, −4, −2, 2, 4 (2034 digits, December 2020, Serge Batalov, NEWPGEN, OPENPFGW, PRIMO)

    126831252923413 * 4657# / 273 + d, d = 1, 3, 7, 9, 13 (2002 digits, 8 Nov 2020, Peter Kaiser, PRIMO)

    394254311495 * 3733# / 2 + d, d = -8, -4, -2, 2, 4 (1606 digits, Nov 2017, Serge Batalov, NEWPGEN, OPENPFGW, PRIMO)

    2316765173284 * 3600# + 16061 + d, d = 0, 2, 6, 8, 12 (1543 digits, 16 Oct 2016, Norman Luhn, PRIMO)

    163252711105 * 3371# / 2 + d, d = −8, −4, −2, 2, 4 (1443 digits, Jan 2014, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

    9039840848561 * 3299# / 35 + d, d = −5, −1, 1, 5, 7 (1401 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

    699549860111847 * 24244 + d, d = −1, 1, 5, 7, 11 (1293 digits, Dec 2013, Reto Keiser, R. Gerbicz, PFGW, PRIMO)

    405095429109490796 * 2683# + 16057 + d, d = 0, 4, 6, 10, 12 (1150 digits, 4 Jul 2020, Michael Bell, RIEMINER, ECPP-DJ)

    566650659276 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)

    554729409262 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)

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    6. The Largest Known Prime Sextuplets

    28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 (1037 digits, 14 Mar 2016, Norman Luhn, APSIEVE, PRIMO)

    6646873760397777881866826327962099685830865900246688640856 * 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16 (780 digits, 8 Nov 2018, Vidar Nakling, PRIMO)

    29720510172503062360713760607985203309940766118866743491802189150471978534404249 * 22299 + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16 (772 digits, 1/28/2018, Riecoin #822096)

    29749903422302373222996698880833194129159047179535887991184960156219652236318921 * 22293 + 679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16 (770 digits, 12/9/2017, Riecoin #793872)

    29696802688480280387313212926526693549449146292085717645262228449092881114972806 * 22290 + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16 (769 digits, 2/25/2018, Riecoin #838224)

    29744205023784420961031622414734790416939049568996819659808238403983863222665068 * 22288 + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16 (769 digits, 2/18/2018, Riecoin #834192)

    29707412718946949415029080194980493978605678414396606766712262274235284928962561 * 22278 + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16 (766 digits, 1/14/2018, Riecoin #814032)

    29696978890366869883141509418765838581871522982358338407613039711378021084519043 * 22259 + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16 (760 digits, 12/31/2017, Riecoin #805968)

    29691575669072177222494655186416928710256802541243921484227880404600991044790342 * 22259 + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16 (760 digits, 12/16/2017, Riecoin #797904)

    29738370152765841200477916368997470863233149039979929714395166089470825913521999 * 22250 + 3267273123746637724423731592929240166353975680818870504129389950929427468581 + d, d = 0, 4, 6, 10, 12, 16 (757 digits, 2/11/2018, Riecoin #830160)

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    7. The Largest Known Prime Septuplets

    113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 (1002 digits, 27 Jan 2021, Peter Kaiser)

    32821868878860201045633341031688415601401701228 32878265333984717524446848642006351778066196724473 92249620201536539259942023218972369026762290403609 01005487309186655777663859063397693729163631275766 07799875309038457637116938538279395260265064447747 74261236889041020217108597484837589978261046949778 71991825164994665583879769659044973939714534960362 41885200541893611077817261813672809971503287259089 * 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 (527 digits, 16 Jun 2019, Vidar Nakling, RIEMINER0.9, PRIMO)

    115828580393941*1200# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20 (515 digits, 18 Jan 2018, Norman Luhn, PRIMO)

    4733578067069 * 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20 (402 digits, May 2016, Norman Luhn)

    687001431518312990252195799540952 * 719# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)

    686636073174158279347746711902518 * 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)

    686488342697495738978150794512038 * 719# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)

    686305940768787196771563962517233 * 719# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)

    686129792256610907998640667932122 * 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)

    685817639451814894948541841801955 * 719# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20 (331 digits, 25 Sep 2020, Michalis Christou, Rieminer)

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    8. The Largest Known Prime Octuplets

    6879356578124627875380298699944709053335 * 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (324 digits, 12 Mar 2021, Michalis Christou)

    6492845263546348546 * 700# + 226449521 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)

    6906530913807107618746553985178874306453389164948020895483633515626362605252139055492044931282748833678120293 * 467# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (305 digits, 22 Aug 2021, Riecoin #1567265)

    359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)

    13445129483076831776879668667126924653719927344389197934344708782916844562826159383097747236770844504282991 * 467# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (302 digits, 22 Aug 2021, Riecoin #1567173)

    857102714401291321747215142362839471354082701279376732604183515709764289908086764114487474405626488688705735862631 * 457# + 114355384736051 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (302 digits, 21 Aug 2021, Riecoin #1566637)

    3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501 * 463# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (302 digits, 22 Aug 2021, Riecoin #1567399)

    99852720003486908985920576060022292734444335215737143674166366804523967920443149768132168626265427179517285486 * 461# + 114521428533971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (301 digits, 21 Aug 2021, Riecoin #1566618)

    128225790098449462554791725702660341231983505261738200778850470659597812508251249240284606022535250058728805 * 463# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (300 digits, 22 Aug 2021, Riecoin #1566997)

    121234078409927149034019206958790573974019543760792101928469116523098424880257116547389250193358233179853793 * 463# + 27899359258013 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (300 digits, 21 Aug 2021, Riecoin #1566147)

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    9. The Largest Known Prime Nonuplets

    3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501 * 463# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (302 digits, 22 Aug 2021, Riecoin #1567399)

    1620259924615470570706663156278905026372754732844252658390408090245313172792664271166384219300680488342402961778 * 450# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (296 digits, 20 Aug 2021, Riecoin #1566093)

    40893595297845006551741048717748959451570266851095389722761855002653709793065456232477944049520841797242 * 457# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (291 digits, 07 Sep 2021, Riecoin #1576463)

    732510298464897302406863094406522970022698347078480858977704780162529246459062655850703116903119380663871601 * 440# + 114521428533971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (288 digits, 20 Aug 2021, Riecoin #1558658)

    115750297122237222935734922477288507624860627469651014827576218647836337473108302145884182747351935638522131 * 439# + 980125031081081d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (287 digits, 24 Aug 2021, Riecoin #1568122)

    1519007803707207636172890273323957030631282142918279301959878723805156323295860440667348707667809976826323916 * 433# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (285 digits, 20 Aug 2021, Riecoin #1538130)

    2013582968688100254021601080203506492283895313221178016484151445253584656191978234310137219527 * 461# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (284 digits, 24 Aug 2021, Riecoin #1568391)

    34054655905291789780022004039187458974596733542616362631849659033442328237809401298372608722059630550 * 443# + 114355384736051 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (283 digits, 20 Aug 2021, Riecoin #1542990)

    2308592721935770706011673286141005968738805658628030547206190428961516998627192255910113817055311300254411 * 433# + 443493962465989 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (283 digits, 20 Aug 2021, Riecoin #1539667)

    14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550 * 449# + 226554621544609 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (282 digits, 12 Sep 2021, Riecoin #1579367)

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    10. The Largest Known Prime Decuplets

    14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550 * 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (282 digits, 12 Sep 2021, Riecoin #1579367, PrimaPoolSolo)

    290901656335108169864195656135043662615199446375386143995339722400236057821426952579658098504166333411889 * 401# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (269 digits, 27 Jul 2021, Riecoin #1551825, tentuple)

    33521646378383216495527 * 331# + 4700094892301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (156 digits, 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 digits, 9 Feb 2017, Norman Luhn)

    7425 * 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (120 digits, May 2016, Roger Thompson)

    118557188915212 * 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (118 digits, Jun 2014, Norman Luhn)

    13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 53586844409797545 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 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 digits, 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 digits, 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 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

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    11. The Largest Known Prime 11-tuplets

    13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (108 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 28 May 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 digits, 23 Sep 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

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    12. The Largest Known Prime Dodecuplets

    13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (108 digits, 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 digits, Sep 2014, Michael Stocker, PRIMO)

    467756 * 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (66 digits, May 2014, Roger Thompson)

    59125383480754 * 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (61 digits, Sep 2013, Michael Stocker)

    78989413043158 * 109# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (59 digits, Jan 2010, Norman Luhn)

    450725899 * 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (56 digits, Nov 2014, Martin Raab)

    14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, Feb 2013, Roger Thompson)

    8486221 * 107# + 4549290807806861 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)

    10000000000000000000000000000929532973818094710897 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (50 digits, 24 Feb 2021, Norman Luhn)

    10000000000000000000000000000896396147387349765031 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, 24 Feb 2021, Norman Luhn)

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    13. The Largest Known Prime 13-tuplets

    4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (61 digits, 23 Mar 2017, Norman Luhn)

    14815550 * 107# + 4385574275277311 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (50 digits, Feb 2013, Roger Thompson)

    14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (50 digits, Feb 2013, Roger Thompson)

    61571 * 107# + 4803194122972361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (48 digits, Aug 2009, Jens Kruse Andersen)

    381955327397348*80# + 18393209 + d, d = 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)

    381955327397348*80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (46 digits, Dec 2007, Norman Luhn)

    1000000000000000027545153594708289884461 + d, d= 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (40 digits, 13 Jul 2021, Norman Luhn)

    1000000000000000014210159036148101380473 + d, d= 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (40 digits, 13 Jul 2021, Norman Luhn)

    1000000000000000014210159036148101380471 + d, d= 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (40 digits, 13 Jul 2021, Norman Luhn)

    1000000000000000005621788289386343008051 + d, d= 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (40 digits, 13 Jul 2021, Norman Luhn)

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    14. The Largest Known Prime 14-tuplets

    14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (50 digits, Feb 2013, Roger Thompson)

    381955327397348*80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)

    1000000000000000014210159036148101380471 + d, d = 0, 2, 6, 8,12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (40 digits, 21 Aug 2021, Norman Luhn)

    1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (40 digits, 10 Mar 2021, Norman Luhn)

    10000000000009283441665311798539399 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (35 digits, Feb 2021, Norman Luhn)

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

    26093748 * 67# + 383123187762431 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, 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 digits, Apr 2008, Jens Kruse Andersen)

    107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

    101885197790002105359911556070661 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

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    15. The Largest Known Prime 15-tuplets

    33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (40 digits, 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 digits, 18 Nov 2016, Roger Thompson)

    10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (35 digits, Sep 2012, Roger Thompson)

    302458608131364933637125192102583 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (33 digits, Feb 2021, Roger Thompson)

    150048143328514263089612453401301 + d, d = 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62 (33 digits, Feb 2021, Roger Thompson)

    107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (33 digits, Apr 2008, Jens Kruse Andersen)

    99999999948164978600250563546400 + d, d = 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67 (32 digits, Nov 2004, Joerg Waldvogel and Peter Leikauf)

    1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (31 digits, 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 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)

    1003234871202624616703163933853 + d, d = 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)

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    16. The Largest Known Prime 16-tuplets

    322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (35 digits, 18 Nov 2016, Roger Thompson)

    302458608131364933637125192102583 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (33 digits, Feb 2021, Roger Thompson)

    150048143328514263089612453401301 + d, d = 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 (33 digits, Feb 2021, Roger Thompson)

    1003234871202624616703163933853 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)

    11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (29 digits, Jan 2012, Roger Thompson)

    5867208169546174917450987997 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

    5621078036155517013724659007 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

    4668263977931056970475231217 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

    4652363394518920290108071167 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

    4483200447126419500533043987 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

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    17. The Largest Known Prime 17-tuplets

    150048143328514263089612453401301 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 (33 digits, 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 digits, 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 digits, 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 digits, Jan 2012, Roger Thompson)

    5867208169546174917450987997 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, 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 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

    5621078036155517013724659007 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, 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 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

    4668263977931056970475231217 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, 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 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

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    18. The Largest Known Prime 18-tuplets

    5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, Feb 2013, Raanan Chermoni & Jaroslaw Wroblewski)

    2406179998282157386567481191 + d, d = 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)

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    19. The Largest Known Prime 19-tuplets

    622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76 (30 digits, December 27, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

    248283957683772055928836513589 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)

    138433730977092118055599751669 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski)

    39433867730216371575457664399 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (29 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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 digits, 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}

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    20. The Largest Known Prime 20-tuplets

    1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (31 digits, May 23, 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 digits, January 21, 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 digits, December 26, 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 digits, December 19, 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 digits, November 17, 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 digits, October 20, 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 digits, September 18, 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 digits, June 19, 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 digits, March 23, 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 digits, October 28, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    More

    21. The Largest Known Prime 21-tuplets

    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 digits, December 27, 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 digits, 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 (30 digits, 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 digits, 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}

    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 Sep 2016
    3 20008 4111286921397 * 266420 + d, d = −1, 1, 5 Peter Kaiser, POLYSIEVE, LLR, PRIMO 24 Apr 2019
    4 10132 667674063382677 * 233608 + d, d = −1, 1, 5, 7 Peter Kaiser, PRIMO 27 Feb 2019
    5 2034 566761969187 * 4733#/2 + d, d = −8, −4, −2, 2, 4 Serge Batalov, NEWPGEN, OPENPFGW, PRIMO December 2020
    6 1037 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 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
    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, PFGW, 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, PFGW, PRIMO 23 Sep 2019
    13 61 4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 Norman Luhn 23 Mar 2017
    14 50 14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 Roger Thompson Feb 2013
    15 40 33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 Norman Luhn 25 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 150048143328514263089612453401301 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 Roger Thompson 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 May 23, 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 Joerg Waldvogel 1996
    16 21 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 Joerg Waldvogel 1996
    17 22 1620784518619319025971 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 Joerg 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 Joerg 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 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 Dec 1998
    8 110 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 Norman Luhn Feb 2001
    9 110 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 Norman Luhn Feb 2001
    10 103 72613488698235 * 227# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 Norman Luhn 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 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 1000 digits
    k Digits Prime k-tuplet Who When
    1 1332 24423 − 1 Alexander Hurwitz Nov 1961
    2 1040 256200945 * 23426 ± 1 Oliver Atkin & N. W. Rickert 1980
    3 1083 437850590*(23567 − 21189) − 6*21189 + d, d = −5, −1, 1 Tony Forbes Dec 1996
    4 1004 76912895956636885*(23279 − 21093) − 6*21093 + d, d = −7, −5, −1, 1 Tony Forbes Sep 1998
    5 1034 31969211688*2400# + 16061 + d, d = 0, 2, 6, 8, 12 Norman Luhn Jul 2002
    6 1037 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 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 10000 digits
    k Digits Prime k-tuplet Who When
    1 13395 244497 − 1 Harry Nelson & David Slowinski Apr 1979
    2 11713 242206083 * 238880 ± 1 H. K. Indlekofer & A. Járai Nov 1995
    3 10047 2072644824759 * 233333 + d, d = −1, 1, 5 Norman Luhn, François Morain, FastECPP Nov 2008
    4 10132 667674063382677 * 233608 + d, d = −1, 1, 5, 7 Peter Kaiser, PRIMO 27 Feb 2019

    Early discovery of 100000 digits
    k Digits Prime k-tuplet Who When
    1 227832 2756839 − 1 David Slowinski & Paul Gage Apr 1992
    2 100355 65516468355 * 2333333 ± 1 Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR Aug 2009

    Early discovery of 1000000 digits
    k Digits Prime k-tuplet Who When
    1 2098960 26972593 − 1 Nayan Hajratwala, George Woltman, Scott Kurowski et al (GIMPS) Jun 1999

    Early discovery of 10000000 digits
    k Digits Prime k-tuplet Who When
    1 12978189 243112609 − 1 Edson Smith, George Woltman, Scott Kurowski et al (GIMPS) Sep 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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