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; anthony.d.forbes@gmail.com.

Site address: http://anthony.d.forbes.googlepages.com/ktuplets.htm.

Recent additions

26 Nov 2019
Prime 9-tuplet (NEW RECORD!)
13508939513048 88395921079397906146942737805199452428593739359909 06783020181836989730867402793793099583190261126801 57777828605797819467408820803534396641509694073174 83144667462762512001500980249401626599331148214351 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (214 digits, 25 Nov 2019, Thomas Nguyen)
Prime 8-tuplets
27628080896395380 73669425900363696791472319312618180989379204670603 85187096436202947820337610267085978774148702787312 68049422491381661071974437691983894573267781856898 01128711013611851960165316892743914885794898364541 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (217 digits, 25 Nov 2019, Thomas Nguyen)
6739366438694 29199449178578320692645616681241034261240729725069 88670364803897819647546530618992702027278326460169 73472126556576261747467880785921296368790408745962 47953797275493231279913426997044449300238293801511 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (213 digits, 25 Nov 2019, Thomas Nguyen)
3375090138339 42551508254862489695685228893113655010051431516962 34731772753194796724824225061927682936654596799204 33251957562063635434640798149693498340936586992225 91865445983962775508718021547812212603749735911121 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (213 digits, 25 Nov 2019, Thomas Nguyen)
1689624146748 70974316568313713792360939033575671110162979521354 81081274818031692021004864440760062948185099858998 90017157942113196339584894087140873512481248076161 39472595468130334987998029820723877434796927059101 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (213 digits, 25 Nov 2019, Thomas Nguyen)
841681705143 09397596111951900931898318489683443148265095464659 91628502633715675025061451345297801663537414422903 70584757155832588970501001969740332807127560088027 50298256601149413019369595594235998494393460088391 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (212 digits, 25 Nov 2019, Thomas Nguyen)
422172469565 41253061516015534601435783825478202548175216338147 68419344568613331021731828272888869632812421223934 23356343284307901259679010610488114476343719304324 03324319797135110345561140440402272083234025148461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (212 digits, 25 Nov 2019, Thomas Nguyen)

29 Oct 2019
Prime 20-tuplet (NEW RECORD!)
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)

19 Oct 2019
Prime octuplet (NEW RECORD!)
180315603 66632494173624649961103800895676665431732439349797 94200800234191227707538758604845380587451507379242 72736595181879630622419937962884400457684262508916 90322717353944842567298670135288556860121687657495 41436837851540527779448535759667215640559772307202 10975532569630677164985160348078053972930223236301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)
Prime septuplets
180443319 97278727895701626588207741775759978458199915815330 73456968619567828320321215894999987978576899740270 83304629680475438469017669444400602229554882688104 26943114779381262762067189091991560135401695298538 14086500199141944657274860445697285170745389433343 52860013609484423134037849199381223834279227983611 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)
180390176 97184376450105059282728120587310104988222663864051 92043013963847321370074225000433608302754008578020 83009621881638863476300447992768515528715888648209 33864458212649903384740100108935783962585698318361 73439473165007809442986166715965444401377209055115 68074149925885913962072969387526117937499619879331 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)
180379632 69447770538975493764322433942734819440053174821861 45630538713594531927572567359498047987561097005374 16195526426016442196127100280518921757565466740558 85006391730079030383533062155049024883381032172380 67542259115254283585531296820038296544789596325888 80895550908145684739226530858777649242979192280251 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)
180359844 70047559946673806327164988311864970157301356260624 65080635886759722592270376966158282163669725626296 53089269059320426640528647488124598144667670585577 46200310120064989464717737240197168954011565741944 70489521123832979631498365092735654079830170290847 91793650691087987312852713243327721304745269037661 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)
180329405 84554032455181768148137406527873286663318416335630 73117671700378999398759599504049414640306214250824 09522094185834538517532819885349269446785851700434 30477594417841827368011683000265759196533422448602 38788399225535466924471428624618961769586569917287 82248655142713551255719773077840290609697581131601 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)
21 more

18 Oct 2019
Prime 20-tuplet (NEW RECORD!)
831504454982803270879178298359 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, October 17, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

27 Sep 2019
Prime 20-tuplet (NEW RECORD!)
807462397198198801670343382679 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, September 25, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

23 Sep 2019
Prime 12-tuplets (NEW RECORD! AND FIRST TO EXCEED 100 DIGITS!)
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)
Prime 11-tuplets (NEW RECORD!)
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 + 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)
Prime 10-tuplets
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)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 * 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 + 48866957363924465 * 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 + 48450891632938904 * 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 + 48246843678337726 * 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 + 47856930537036851 * 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 + 46622982649030457 * 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)
... 45 more

03 Sep 2019
Prime 20-tuplet (NEW RECORD!)
789292095021856634277511882469 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, August 19, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

18 Jul 2019
Prime 20-tuplet (NEW WORLD RECORD)
764364269069907627842423582909 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, July 17, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

18 Jul 2019
Prime 20-tuplet (NEW WORLD RECORD)
764364269069907627842423582909 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, July 17, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

16 Jun 2019
Prime 7-tuplet (NEW WORLD RECORD!)
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)

29 May 2019
Prime 11-tuplet (NEW WORLD RECORD!)
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)

Prime 10-tuplets
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 9268026349694711 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4265997347677925 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (107 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

Smallest 400-digit prime sextuplet
10^399 + 33756090918084087 + d, d = 0, 4, 6, 10, 12, 16 (400 digits, 7 May 2019, Norman Luhn, PRIMO)

Smallest titanic prime quintuplet to pattern d = 0, 4, 6, 10, 12
10^999 + 3818999670116007 + d, d = 0, 4, 6, 10, 12 (1000 digits, 9 May 2019, Norman Luhn, PRIMO)

Prime triplets
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)

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. Odds and Ends
  24. Links to Related Material
  25. Mathematical Background
  26. References
  27. 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 50 digits, found by G. John Stevens in 1995 [S95]:

    10000000000000000000000000000000000000000058537891,
    10000000000000000000000000000000000000000058537893,
    10000000000000000000000000000000000000000058537897,
    10000000000000000000000000000000000000000058537899.

    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)

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

    70965694293 * 2200003 ± 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)

    38529154785 * 2173250 ± 1 (52165 digits, Jul 2014, Serge Batalov, NEWPGEN, LLR)

    194772106074315 * 2171960 ± 1 (51780 digits, Jun 2007, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)

    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)

    3221449497221499 * 234567 + d, d = −1, 1, 5 (10422 digits, Sep 2015, Peter Kaiser, NEWGEN, LLR, PRIMO5)

    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)

    667674063382677 * 233608 + d, d = −1, 1, 5 (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)

    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)

    25796119248 * 4987# / 35 + d, d = −1, 1, 5, 7 (2135 digits, May 2011, Gary Chaffey)

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    5. The Largest Known Prime Quintuplets

    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)

    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)

    424232794973 * 2600# + 43777 + d, d = 0, 4, 6, 10, 12 (1107 digits, Mar 2009, Norman Luhn, PRIMO)

    283534892623 * 2500# + 1091261 + d, d = 0, 2, 6, 8, 12 (1069 digits, Apr 2006, Norman Luhn)

    96972480423104 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12 (1038 digits, Nov 2012, Norman Luhn, PRIMO)

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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

    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)

    10^319 + 2219844666811981651 + d, d = 0, 2, 6, 8, 12, 18, 20 (320 digits, 9 Apr 2019, Norman Luhn, PRIMO)

    4079068497377 * 739# / 14 + d, d = −4, −2, 2, 4, 8, 14, 16 (319 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

    180443319 97278727895701626588207741775759978458199915815330 73456968619567828320321215894999987978576899740270 83304629680475438469017669444400602229554882688104 26943114779381262762067189091991560135401695298538 14086500199141944657274860445697285170745389433343 52860013609484423134037849199381223834279227983611 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)

    180390176 97184376450105059282728120587310104988222663864051 92043013963847321370074225000433608302754008578020 83009621881638863476300447992768515528715888648209 33864458212649903384740100108935783962585698318361 73439473165007809442986166715965444401377209055115 68074149925885913962072969387526117937499619879331 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)

    180379632 69447770538975493764322433942734819440053174821861 45630538713594531927572567359498047987561097005374 16195526426016442196127100280518921757565466740558 85006391730079030383533062155049024883381032172380 67542259115254283585531296820038296544789596325888 80895550908145684739226530858777649242979192280251 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)

    180359844 70047559946673806327164988311864970157301356260624 65080635886759722592270376966158282163669725626296 53089269059320426640528647488124598144667670585577 46200310120064989464717737240197168954011565741944 70489521123832979631498365092735654079830170290847 91793650691087987312852713243327721304745269037661 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)

    180329405 84554032455181768148137406527873286663318416335630 73117671700378999398759599504049414640306214250824 09522094185834538517532819885349269446785851700434 30477594417841827368011683000265759196533422448602 38788399225535466924471428624618961769586569917287 82248655142713551255719773077840290609697581131601 + d, d = 0, 2, 6, 8, 12, 18, 20 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)

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    8. The Largest Known Prime Octuplets

    180315603 66632494173624649961103800895676665431732439349797 94200800234191227707538758604845380587451507379242 72736595181879630622419937962884400457684262508916 90322717353944842567298670135288556860121687657495 41436837851540527779448535759667215640559772307202 10975532569630677164985160348078053972930223236301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (309 digits, 19 Oct 2019, Thomas Nguyen, RIEMINER 0.91)

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

    29995576270632 * 550# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (236 digits, Jun 2014, Norman Luhn)

    330846961 * 503# + 349129635971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (218 digits, Feb 2008, Jens Kruse Andersen)

    27628080896395380 73669425900363696791472319312618180989379204670603 85187096436202947820337610267085978774148702787312 68049422491381661071974437691983894573267781856898 01128711013611851960165316892743914885794898364541 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (217 digits, 25 Nov 2019, Thomas Nguyen)

    6739366438694 29199449178578320692645616681241034261240729725069 88670364803897819647546530618992702027278326460169 73472126556576261747467880785921296368790408745962 47953797275493231279913426997044449300238293801511 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (213 digits, 25 Nov 2019, Thomas Nguyen)

    3375090138339 42551508254862489695685228893113655010051431516962 34731772753194796724824225061927682936654596799204 33251957562063635434640798149693498340936586992225 91865445983962775508718021547812212603749735911121 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (213 digits, 25 Nov 2019, Thomas Nguyen)

    1689624146748 70974316568313713792360939033575671110162979521354 81081274818031692021004864440760062948185099858998 90017157942113196339584894087140873512481248076161 39472595468130334987998029820723877434796927059101 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (213 digits, 25 Nov 2019, Thomas Nguyen)

    841681705143 09397596111951900931898318489683443148265095464659 91628502633715675025061451345297801663537414422903 70584757155832588970501001969740332807127560088027 50298256601149413019369595594235998494393460088391 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (212 digits, 25 Nov 2019, Thomas Nguyen)

    422172469565 41253061516015534601435783825478202548175216338147 68419344568613331021731828272888869632812421223934 23356343284307901259679010610488114476343719304324 03324319797135110345561140440402272083234025148461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (212 digits, 25 Nov 2019, Thomas Nguyen)

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    9. The Largest Known Prime Nonuplets

    13508939513048 88395921079397906146942737805199452428593739359909 06783020181836989730867402793793099583190261126801 57777828605797819467408820803534396641509694073174 83144667462762512001500980249401626599331148214351 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (214 digits, 25 Nov 2019, Thomas Nguyen)

    663579549486449 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (203 digits, 16 Mar 2017, Norman Luhn)

    68663510211259 * 337# + 88789 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (150 digits, Jan 2010, Norman Luhn)

    3336884 * 331# + 80877403191701 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (140 digits, Sep 2007, Dirk Augustin & Jens Kruse Andersen)

    851437873414817 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (136 digits, 9 Feb 2017, Norman Luhn)

    772556746441918 * 300# + 29247919 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (136 digits, 9 Feb 2017, Norman Luhn)

    772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (136 digits, 9 Feb 2017, Norman Luhn)

    394833958615791 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (135 digits, 9 Feb 2017, Norman Luhn)

    106345403186416 * 300# + 29247913 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30 (135 digits, 9 Feb 2017, Norman Luhn)

    90421624808713 * 300# + 103498931 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (135 digits, Feb 2005, Norman Luhn)

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    10. The Largest Known Prime Decuplets

    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)

    13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 * 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 + 48866957363924465 * 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 + 48450891632938904 * 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)

    839858 * 107# + 2566964683459061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (49 digits, Aug 2009, Jens Kruse Andersen)

    337712 * 107# + 3440354553191441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (48 digits, Aug 2009, Jens Kruse Andersen)

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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)

    1955206838 * 73# + 44208109063 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (38 digits, Aug 2012, Martin Raab)

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

    1464893944 * 67# + 42166182984041 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (35 digits, Jul 2012, Martin Raab)

    457308940 * 67# + 4122369405991 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 (34 digits, Mar 2011, Martin Raab)

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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)

    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)

    101803109763079694387921584406441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

    101047123513223569167212934432341 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

    100859765410802682029505696121301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

    100496797396678760339871075201851 + 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)

    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)

    999999999962618227626700812281 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (30 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)

    100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (30 digits, Feb 2013, 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)

    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)

    3361885098594416802447362317 + d, d = 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 = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)

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    17. The Largest Known Prime 17-tuplets

    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)

    4652363394518920290108071167 + 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)

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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}

    20. The Largest Known Prime 20-tuplets

    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)

    831504454982803270879178298359 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, October 17, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    807462397198198801670343382679 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, September 25, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    789292095021856634277511882469 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, August 19, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    764364269069907627842423582909 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, July 17, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    701870455949526598513130862539 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, April 7, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    686962597479437604159786541481 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, April 27, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    667424014858149638371951648871 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, February 18, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

    639121700726230052098229452019 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, December 23, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

    622803914376064301858782434517 + d, d = 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)

    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 1606 394254311495 * 3733# / 2 + d, d = -8, -4, -2, 2, 4 Serge Batalov, NEWPGEN, OPENPFGW, PRIMO Nov 2017
    6 1037 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 Norman Luhn, APSIEVE, PRIMO 14 Mar 2016
    7 527 32821868878860201045633341031688415601401701228 32878265333984717524446848642006351778066196724473 92249620201536539259942023218972369026762290403609 01005487309186655777663859063397693729163631275766 07799875309038457637116938538279395260265064447747 74261236889041020217108597484837589978261046949778 71991825164994665583879769659044973939714534960362 41885200541893611077817261813672809971503287259089 * 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 Vidar Nakling, RIEMINER0.9, PRIMO 16 Jun 2019
    8 309 180315603 66632494173624649961103800895676665431732439349797 94200800234191227707538758604845380587451507379242 72736595181879630622419937962884400457684262508916 90322717353944842567298670135288556860121687657495 41436837851540527779448535759667215640559772307202 10975532569630677164985160348078053972930223236301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 Thomas Nguyen, RIEMINER 0.91 19 Oct 2019
    9 203 663579549486449 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 Norman Luhn Mar 2017
    10 136 772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 Norman Luhn 9 Feb 2017
    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 30 100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 Roger Thompson Feb 2013
    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 30 839013472011818416634745523991 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 Raanan Chermoni & Jaroslaw Wroblewski 28 Oct 2019
    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 - -

    First appearance 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

    First appearance 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

    First appearance 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

    First appearance 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

    First appearance 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

    First appearance 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

    First appearance 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. Odds and Ends

    List of all possible patterns of prime k-tuplets

    List of the smallest prime k-tuplets

    Various lists of prime k-tuplets

    Near misses: Clusters of primes that didn't quite make it into the main list

    The Hardy-Littlewood constants pertaining to the distribution of prime k-tuplets [HL22]

    Site History

    24. Links to Related Material

    Jens Kruse Andersen: The Largest Known Simultaneous Primes

    Jens Kruse Andersen: Consecutive Primes in Arithmetic Progression

    Jens Kruse Andersen: Largest Consecutive Factorizations

    Dirk Augustin: Cunningham Chain records

    Chris K. Caldwell: The Largest Known Primes

    Chris K. Caldwell: Top twenty twin primes

    TF: Ten consecutive primes in arithmetic progression

    Dr. Minh. L. Perez Press: Smarandache Primes

    N. J. A. Sloane: On-Line Encyclopedia of Integer Sequences

    Manfred Toplic: The Nine and Ten Primes Project

    Robin Whitty: Theorem of the Day

    25. 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).

    26. 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

    [R95] Warut Roonguthai, Prime quadruplets, NMBRTHRY Mailing List, September 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.

    [S95] G. John Stevens, Prime Quadruplets, J. Recr. Math. 27 (1995), 17-22.