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.
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Contents
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, interdecal 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 tenblock. Hence there must be exactly one with each of these unit digits. And just to illustrate the point, here is another example; the smallest proven prime quadruplet of 2000 digits, found by Gerd Lamprecht in Oct 2017:
Prime ktuplets
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 11tuplets. I couldn't find the general term 'ktuplets' 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 ktuplet 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 ktuplets for k = 2, 3, 4, ..., 20 and 21. I do not know of any prime ktuplets 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 + b_{1}, b_{2}, ..., b_{k} is used (only in linked pages) to denote the k numbers {N + b_{1}, N + b_{2}, ..., N + b_{k}}.
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 ktuplets 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 doublecheck them myself. Otherwise I would appreciate some indication of how you proved that your numbers are true primes. Email address: pzktupel[at]pzktupel[dot]de.
Overview of largest known & early discovery of a nontrivial prime ktuplet to given pattern. 
Prime Counting Functions • Tables of values of π(x) up to π_{21}(x) 
Tables of values of π_{k}(10^{n}) n=1..17, k=1..16 
Record Gap Tables of prime ktuplets / First Occurrence Gap 
Initial members of prime ktuplets 
The smallest ndigit prime ktuplets 
First initial members of consecutive prime ktuplets (PART I) 
First initial members of consecutive prime ktuplets (PART II) 
Possible patterns & the HardyLittlewood constants of prime ktuplets [HL22]. 
Rank  The Largest Known Twin Primes  Digits  When  Additions 
1  2996863034895 • 2^{1290000} ± 1  388342  19 Sep 2016  Tom Greer, TwinGen, PrimeGrid, LLR 
2  3756801695685 • 2^{666669} ± 1  200700  26 Dec 2011  Timothy Winslow, TwinGen, PrimeGrid, LLR 
3  669821552^{16384} − 669821552^{8192} ± 1  144605  26 Apr 2024  Roman Trunov; Cyclo; PRST 
4  222710306^{16384} − 222710306^{8192} ± 1  136770  29 Mar 2024  Bruce Marler; Cyclo; PRST 
5  65516468355 • 2^{333333} ± 1  100355  15 Aug 2009  Peter Kaiser, NewPGen, PrimeGrid, TPS, LLR 
6  201926367 • 2^{266668} ± 1  80284  6 May 2024  Göran Schmidt, NewPGen, PRST 
7  160204065 • 2^{262148} ± 1  78923  8 Jul 2021  Erwin Doescher, LLR 
8  1893611985^{8192} − 1893611985^{4096} ± 1  76000  06 Mar 2024  Bruce Marler; Cyclo; PRST 
9  1589173270^{8192} − 1589173270^{4096} ± 1  75376  24 Feb 2024  Frank Doornink; Cyclo; PRST 
10  996094234^{8192} − 996094234^{4096} ± 1  73715  26 Jan 2024  Roman Trunov; Cyclo; PRST 
More Twin Primes 
Rank  The Largest Known Primes Triplets  Digits  When  Additions  Certificates 
1  4111286921397 • 2^{66420} − 1 + d, d = 0, 2, 6  20008  24 Apr 2019  Peter Kaiser, Polysieve, LLR, Primo  click 
2  6521953289619 • 2^{55555} − 5 + d, d = 0, 4, 6  16737  30 Apr 2013  Peter Kaiser  click 
3  56667641271 • 2^{44441} − 1 + d, d = 0, 2, 6  13389  1 Apr 2022  Stephan Schöler, NewPGen, OpenPFGW; Oliver Kruse, Primo  click 
4  4207993863 • 2^{38624 } − 1 + d, d = 0, 2, 6  11637  5 Jun 2021  Frank Doornink, NewPGen, LLR, Primo  click 
5  14059969053 • 2^{36672} − 5 + d, d = 0, 4, 6  11050  17 Jun 2018  Serge Batalov, NewPGen, OpenPFGW, Primo  click 
6  3221449497221499 • 2^{34567} − 1 + d, d = 0, 2, 6  10422  2 Sep 2015  Peter Kaiser, NewPGen, LLR, OpenPFGW, Primo  click 
7  1288726869465789 • 2^{34567} − 5 + d, d = 0, 4, 6  10421  23 Apr 2014  Peter Kaiser, Primo  click 
8  647935598824239 • 2^{33619} − 1 + d, d = 0, 2, 6  10136  22 May 2019  Peter Kaiser, Primo  click 
9  209102639346537 • 2^{33620} − 1 + d, d = 0, 2, 6  10135  22 May 2019  Peter Kaiser, Primo  click 
10  185353103135997 • 2^{33620} − 1 + d, d = 0, 2, 6  10135  22 May 2019  Peter Kaiser, Primo  click 
More Prime Triplets 
Rank  The Largest Known Prime Quadruplets  Digits  When  Additions  Certificates 
1  667674063382677 • 2^{33608} − 1 + d, d = 0, 2, 6, 8  10132  27 Feb 2019  Peter Kaiser, Primo  click 
2  4122429552750669 • 2^{16567} − 1 + d, d = 0, 2, 6, 8  5003  10 Mar 2016  Peter Kaiser, GSIEVE, NewPGen, LLR, Primo  click 
3  (1049713153083 • 2917# • (567 • 2917# + 1) + 11#) • (567 • 2917# − 1) / 7# + 1 + d, d = 0, 2, 6, 8  3753  22 Jul 2023  Ken Davis, APTreeSieve, OpenPFGW, Primo  click 
4  101406820312263 • 2^{12042} − 1 + d, d = 0, 2, 6, 8  3640  13 Jun 2018  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
5  2673092556681 • 15^{3048} − 4 + d, d = 0, 2, 6, 8  3598  14 Sep 2015  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
6  2339662057597 • 10^{3490} + 1 + d, d = 0, 2, 6, 8  3503  21 Dec 2013  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
7  305136484659 • 2^{11399} − 1 + d, d = 0, 2, 6, 8  3443  28 Sep 2013  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
8  722047383902589 • 2^{11111} − 1 + d, d = 0, 2, 6, 8  3360  20 Apr 2013  Reto Keiser, NewPGen, OpenPFGW, Primo  click 
9  585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8  3344  6 Mar 2022  Peter Kaiser, OpenPFGW, Primo  click 
10  43697976428649 • 2^{9999} − 1 + d, d = 0, 2, 6, 8  3024  24 Mar 2012  Peter Kaiser, Primo 3.0.9  click 
More Prime Quadruplets 
Rank  The Largest Known Prime Quintuplets  Digits  When  Additions  Certificates 
1  585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8, 12  3344  6 Mar 2022  Peter Kaiser, OpenPFGW, Primo  click 
2  566761969187 • 4733# / 2 − 8 + d, d = 0, 4, 6, 10, 12  2034  6 Dec 2020  Serge Batalov, NewPGen, OpenPFGW, Primo  click 
3  126831252923413 • 4657# / 273 + 1 + d, d = 0, 2, 6, 8, 12  2002  8 Nov 2020  Peter Kaiser, Primo  click 
4  394254311495 • 3733# / 2 − 8 + d, d = 0, 4, 6, 10, 12  1606  30 Nov 2017  Serge Batalov, NewPGen, OpenPFGW, Primo  click 
5  2316765173284 • 3600# + 16061 + d, d = 0, 2, 6, 8, 12  1543  16 Oct 2016  Norman Luhn, Primo  click 
6  163252711105 • 3371# / 2 − 8 + d, d = 0, 4, 6, 10, 12  1443  1 Jan 2014  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
7  9039840848561 • 3299# / 35 − 5 + d, d = 0, 4, 6, 10, 12  1401  28 Dec 2013  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
8  699549860111847 • 2^{4244} − 1 + d, d = 0, 2, 6, 8, 12  1293  3 Dec 2013  Reto Keiser, R. Gerbicz, OpenPFGW, Primo  click 
9  10^{1199} + 20483870459152351 + d, d = 0, 2, 6, 8, 12  1200  3 Mar 2023  Norman Luhn, OpenPFGW, Primo 3.0.9  click 
10  10^{1199} + 7033048489975137 + d, d = 0, 4, 6, 10, 12  1200  17 Mar 2023  Norman Luhn, OpenPFGW, Primo 3.0.9  click 
More Prime Quintuplets 
Rank  The Largest Known Prime Sextuplets  Digits  When  Additions  Certificates 
1  2^{3700} + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16  1114  8 Nov 2021  Vidar Nakling, Primo, Sixfinder ( based on Riecoin miners ) 
click 
2  28993093368077 • 2399# + 19417 + d, d = 0, 4, 6, 10, 12, 16  1037  14 Mar 2016  Norman Luhn, APSIEVE, Primo  click 
3  6646873760397777881866826327962099685830865900246688640856 • 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16  780  8 Nov 2018  Vidar Nakling, Primo   
4  29720510172503062360713760607985203309940766118866743491802189150471978534404249 • 2^{2299} + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16 
772  28 Jan 2018  Riecoin #822096   
5  29749903422302373222996698880833194129159047179535887991184960156219652236318921 • 2^{2293} + 679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16 
770  28 Sep 2017  Riecoin #793872   
6  29696802688480280387313212926526693549449146292085717645262228449092881114972806 • 2^{2290} + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16 
769  25 Feb 2018  Riecoin #838224   
7  29744205023784420961031622414734790416939049568996819659808238403983863222665068 • 2^{2288} + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16 
769  18 Feb 2018  Riecoin #834192   
8  29707412718946949415029080194980493978605678414396606766712262274235284928962561 • 2^{2278} + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16 
766  14 Jan 2018  Riecoin #814032   
9  29696978890366869883141509418765838581871522982358338407613039711378021084519043 • 2^{2259} + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16 
766  31 Dec 2017  Riecoin #805968   
10  29691575669072177222494655186416928710256802541243921484227880404600991044790342 • 2^{2259} + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16 
760  16 Dec 2017  Riecoin #797904   
More Prime Sextuplets 
Rank  The Largest Known Prime Septuplets  Digits  When  Additions  Certificates 
1  113225039190926127209 • 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20  1002  27 Jan 2021  Peter Kaiser  click 
2  3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473 9224962020153653925994202321897236902676229040360901005487309186655777663859063397693729163631275766 0779987530903845763711693853827939526026506444774774261236889041020217108597484837589978261046949778 7199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089 • 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 
527  16 Jun 2019  Vidar Nakling,
rieMiner 0.9, Primo 
 
3  115828580393941 • 1193# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20  515  18 Jan 2018  Norman Luhn, Primo   
4  1749454900366668261124366444904767913042 • 1051# + 235367427477641 + d, d = 0, 2, 6, 8, 12, 18, 20  482  10 Sep 2023  Michalis Christou, rieMiner   
5  1749440332764626179112277073423987083854 • 1051# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20  482  10 Sep 2023  Michalis Christou, rieMiner   
6  1785036139318226774335532641427159030322 • 991# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20  452  10 Sep 2023  Michalis Christou, rieMiner   
7  4733578067069 • 937# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20  402  9 May 2016  Norman Luhn, Primo   
8  362084778371560960224893228648902482898 • 863# + 235367427477641 + d, d = 0, 2, 6, 8, 12, 18, 20  401  10 Sep 2023  Michalis Christou, rieMiner   
9  362084724493692554397476622571818362892 • 863# + 235290683530361 + d, d = 0, 2, 6, 8, 12, 18, 20  401  10 Sep 2023  Michalis Christou, rieMiner   
10  362084689974572819018431260315189393115 • 863# + 220452326319761 + d, d = 0, 2, 6, 8, 12, 18, 20  401  10 Sep 2023  Michalis Christou, rieMiner   
More Prime Septuplets 
Rank  The Largest Known Prime Octuplets  Digits  When  Additions 
1  362079385668757696008683096558661746463 • 863# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  401  10 Sep 2023  Michalis Christou, rieMiner 0.93a 
2  17823192282008874449172703428792123231110 • 771# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  362  12 Jan 2023  Michalis Christou, rieMiner 
3  531258360785860208657753 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  333  30 Sep 2022  Peter Kaiser, Primo 
4  530956818040688210255681 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  333  30 Sep 2022  Peter Kaiser, Primo 
5  697723422149271424870176724491962624555 • 701# + 145888993435301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  332  28 Apr 2022  Michalis Christou, rieMiner 0.91 
6  5586218959960365309179 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  331  12 Sep 2022  Peter Kaiser, Primo 
7  4869586684665128135306 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  331  30 Sep 2022  Peter Kaiser, Primo 
8  6879356578124627875380298699944709053335 • 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  324  12 Mar 2021  Michalis Christou, rieMiner 0.91 
9  85942978608490853163266464829675186732716531436220205198524648761309585030760262728948076619827920 • 541# + 301570107719123 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
318  7 May 2023  Riecoin #1927251 
10  12535667484694596950800617890716044333306550769988948320664333022424209836994030121791042799528403451 • 523# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 
318  11 May 2023  Riecoin #1929345 
More Prime Octuplets  
Prime Octuplets, found by Riecoin 
Rank  The Largest Known Prime Nonuplets  Digits  When  Additions 
1  182075127245948453356763852678412657384571384320476086323955359028566228121357180020362596219 • 541# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
312  22 Apr 2023  Riecoin #1918654 
2  6981459541055817191260362842479625063402912945070015867718881817316331990854697141515826226327285164890 • 503# + 301713410008249 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
312  7 May 2023  Riecoin #1926945 
3  7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341 • 503# + 220469307413891 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
309  16 Feb 2022  Riecoin #1669866 
4  4544802941746849322755400504979525801125583245162538446615359445516948017861576296257331705348 • 521# + 301713410008249 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
308  27 May 2023  Riecoin #1938898 
5  154140554027854736762047378850909920571712371864836723122728277535436255651743244125969075784222257 • 503# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
307  27 Apr 2023  Riecoin #1921294 
6  8096703388538745581027194751126786700937232194317894651023717668314141648189594312628024001432 • 509# + 145844141558231 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
306  14 Sep 2023  Riecoin #2002325 
7  785736482192035162210293617854084866892633779128067177293795495786556268126995707998914641524 • 509# + 301856712297379 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
305  20 Aug 2023  Riecoin #1987876 
8  121133517275115547018555742318180482744008312012716527913198477819170362242224836572439329489655 • 503# + 226554621544609 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
304  12 Jun 2023  Riecoin #1947664 
9  8765572177843184814229154326623059563840136253586624568046789391874155387479145540897589092413205411 • 491# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
304  29 Jun 2023  Riecoin #1957788 
10  7250929313692442766966693303696774251366194754052517547936232541331423429106473060634375035985565 • 491# + 443216221842829 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
303  10 Apr 2024  Riecoin #2122977 
More Prime Nonuplets  
Prime Nonuplets, found by Riecoin 
Rank  The Largest Known Prime 10tuplets  Digits  When  Additions  
1  14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550 • 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 
282  12 Sep 2021  Riecoin #1579367  
2  290901656335108169864195656135043662615199446375386143995339722400236057821426952579658098504166333411889 • 401# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
269  27 Jul 2021  Riecoin #1551825  
3  14257429881902877844339877915045298096140599288873476083093543949692946630381247693511330479634493 • 367# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
246  18 Aug 2022  Riecoin #1775788  
4  33521646378383216495527 • 331# + 4700094892301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  156  4 Apr 2020  Thomas Nguyen, rieMiner 0.91, MPZAPRCL 

5  772556746441918 • 293# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32  136  9 Feb 2017  Norman Luhn  
6  7425 • 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  120  27 May 2016  Roger Thompson  
7  118557188915212 • 257# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  118  27 Jun 2014  Norman Luhn  
8  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 53586844409797545 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

9  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 51143234991402697 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

10  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 50679161987995696 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

More Prime 10tuplets 
Rank  The Largest Known Prime 11tuplets  Digits  When  Additions  
1  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

2  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 46622982649030457 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

3  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 30796489110940369 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

4  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

5  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20731977215353082 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

6  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20118509988610513 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

7  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 15866045335517629 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

8  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 5238271627884665 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
107  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

9  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
107  28 May 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

10  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 1296173254392493 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
107  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

More Prime 11tuplets 
Rank  The Largest Known Prime 12tuplets  Digits  When  Additions  
1  13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

2  613176722801194 • 151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42  75  30 Sep 2014  Michael Stocker, Primo 

3  467756 • 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  66  20 May 2014  Roger Thompson 

4  9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  66  1 Oct 2021  Roger Thompson 

5  9985397181 • 139# + 249386599747880711 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  66  1 Oct 2021  Roger Thompson 

6  59125383480754 • 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42  61  9 Sep 2013  Michael Stocker 

7  78989413043158 • 109# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  59  18 Jan 2010  Norman Luhn 

8  450725899 • 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  56  4 Nov 2014  Martin Raab 

9  1000000000000000000000000000000002955087732304487826931 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  55  22 Sep 2022  Norman Luhn 

10  1000000000000000000000000000000002760339313453283246757 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42  55  4 Oct 2022  Norman Luhn 

More Prime 12tuplets 
Rank  The Largest Known Prime 13tuplets  Digits  When  Additions  
1  9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  66  1 Oct 2021  Roger Thompson  
2  4135997219394611 • 109# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48  61  23 Mar 2017  Norman Luhn 

3  14815550 • 107# + 4385574275277313 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48  50  5 Feb 2013  Roger Thompson 

4  14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  50  5 Feb 2013  Roger Thompson 

5  10000000000000000000000000019294427203099948114321 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  50  7 May 2023  Norman Luhn 

6  61571 • 107# + 4803194122972361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  48  7 Aug 2009  Jens Kruse Andersen 

7  381955327397348 • 79# + 18393211 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48  46  28 Dec 2007  Norman Luhn 

8  381955327397348 • 79# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48  46  28 Dec 2007  Norman Luhn 

9  100000000000000000000006149198224095343810309 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48  45  2 Mar 2022  Norman Luhn 

10  100000000000000000000004356680452416578030761 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  45  5 Feb 2022  Norman Luhn 

More Prime 13tuplets 
Rank  The Largest Known Prime 14tuplets  Digits  When  Additions  
1  14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  50  5 Feb 2013  Roger Thompson  
2  381955327397348 • 79# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50  46  28 Dec 2007  Norman Luhn 

3  1000000000000000014210159036148101380471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  40  10 Mar 2021  Norman Luhn 

4  1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50  40  10 Mar 2021  Norman Luhn 

5  10000000000009283441665311798539399 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50  35  18 Feb 2021  Norman Luhn 

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

7  26093748 • 67# + 383123187762431 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  33  8 Feb 2005  Christ van Willegen & Jens Kruse Andersen 

8  108804167016152508211944400342691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  33  14 Apr 2008  Jens Kruse Andersen 

9  107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  33  14 Apr 2008  Jens Kruse Andersen 

10  101885197790002105359911556070661 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  33  14 Apr 2008  Jens Kruse Andersen 

More Prime 14tuplets 
Rank  The Largest Known Prime 15tuplets  Digits  When  Additions  
1  33554294028531569 • 61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  40  25 Jan 2017  Norman Luhn  
2  322255 • 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  35  18 Nov 2016  Roger Thompson  
3  10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  35  4 Sep 2012  Roger Thompson  
4  94 • 79# + 1341680294611244014367 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  33  5 Feb 2021  Roger Thompson  
5  3684 • 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  33  5 Feb 2021  Roger Thompson  
6  107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  33  14 Apr 2008  Jens Kruse Andersen  
7  99999999948164978600250563546411 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  32  29 Nov 2004  Jörg Waldvogel and Peter Leikauf  
8  1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  31  16 Oct 2003  Hans Rosenthal & Jens Kruse Andersen  
9  1100916249233879857334075234831 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  31  16 Oct 2003  Hans Rosenthal & Jens Kruse Andersen  
10  1003234871202624616703163933857 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  31  9 Aug 2012  Roger Thompson  
More Prime 15tuplets 
Rank  The Largest Known Prime 16tuplets  Digits  When  Additions  
1  322255 • 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  35  18 Nov 2016  Roger Thompson  
2  94 • 79# + 1341680294611244014363 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  33  5 Feb 2021  Roger Thompson  
3  3684 • 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  33  5 Feb 2021  Roger Thompson  
4  1003234871202624616703163933853 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  31  9 Aug 2012  Roger Thompson  
5  11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  29  2 Jan 2012  Roger Thompson  
6  5867208169546174917450988007 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
7  5621078036155517013724659017 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
8  4668263977931056970475231227 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
9  4652363394518920290108071177 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
10  4483200447126419500533043997 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 16tuplets 
Rank  The Largest Known Prime 17tuplets  Digits  When  Additions  
1  3684 • 73# + 880858118723497737821 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  33  5 Feb 2021  Roger Thompson  
2  100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66  30  19 Feb 2013  Roger Thompson  
3  11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66  29  2 Jan 2012  Roger Thompson  
4  11410793439953412180643704677 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66  29  2 Jan 2012  Roger Thompson  
5  5867208169546174917450988001 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
6  5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
7  5621078036155517013724659011 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
8  5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
9  4668263977931056970475231221 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
10  4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 17tuplets 
Rank  The Largest Known Prime 18tuplets  Digits  When  Additions  
1  5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
2  5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
3  4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4  4652363394518920290108071167 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
5  4483200447126419500533043987 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
6  3361885098594416802447362317 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  30 Jul 2013  Raanan Chermoni & Jaroslaw Wroblewski  
7  3261917553005305074228431077 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  30 Jul 2013  Raanan Chermoni & Jaroslaw Wroblewski  
8  3176488693054534709318830357 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  30 Jul 2013  Raanan Chermoni & Jaroslaw Wroblewski  
9  2650778861583720495199114537 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  25 Feb 2013  Raanan Chermoni & Jaroslaw Wroblewski  
10  2406179998282157386567481197 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  31 Dec 2012  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 18tuplets 
Rank  The Largest Known Prime 19tuplets  Digits  When  Additions 
1  622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76  30  27 Dec 2018  Raanan Chermoni & Jaroslaw Wroblewski 
2  248283957683772055928836513597 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  30  1 Aug 2016  Raanan Chermoni & Jaroslaw Wroblewski 
3  138433730977092118055599751677 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  30  8 Oct 2015  Raanan Chermoni & Jaroslaw Wroblewski 
4  39433867730216371575457664407 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  29  8 Jan 2015  Raanan Chermoni & Jaroslaw Wroblewski 
5  2406179998282157386567481191 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  28  31 Dec 2012  Raanan Chermoni & Jaroslaw Wroblewski 
6  2348190884512663974906615481 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  28  17 Dec 2012  Raanan Chermoni & Jaroslaw Wroblewski 
7  917810189564189435979968491 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  27  29 May 2011  Raanan Chermoni & Jaroslaw Wroblewski 
8  656632460108426841186109951 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  27  19 Feb 2011  Raanan Chermoni & Jaroslaw Wroblewski 
9  630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  27  9 Feb 2011  Raanan Chermoni & Jaroslaw Wroblewski 
10  37 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  23     
More Prime 19tuplets 
Rank  The Largest Known Prime 20tuplets  Digits  When  Additions  
1  1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  23 May 2021  Raanan Chermoni & Jaroslaw Wroblewski  
2  1188350591359110800209379560799 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  21 Jan 2021  Raanan Chermoni & Jaroslaw Wroblewski  
3  1153897621507935436463788957529 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  26 Dec 2020  Raanan Chermoni & Jaroslaw Wroblewski  
4  1135540756371356698957890225821 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  31  19 Dec 2020  Raanan Chermoni & Jaroslaw Wroblewski  
5  1126002593922465663847897293731 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  31  17 Nov 2020  Raanan Chermoni & Jaroslaw Wroblewski  
6  1094372814043722195189448411199 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  20 Oct 2020  Raanan Chermoni & Jaroslaw Wroblewski  
7  1060475118776959297139870952701 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  31  18 Sep 2020  Raanan Chermoni & Jaroslaw Wroblewski  
8  999627565307688186459783232931 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  30  19 Jun 2020  Raanan Chermoni & Jaroslaw Wroblewski  
9  957278727962618711849051282459 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  30  23 Mar 2020  Raanan Chermoni & Jaroslaw Wroblewski  
10  839013472011818416634745523991 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  30  28 Oct 2020  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 20tuplets 
Rank  The Largest Known Prime 21tuplets  Digits  When  Additions  
1  622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84  30  27 Dec 2018  Raanan Chermoni & Jaroslaw Wroblewski  
2  248283957683772055928836513589 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  30  1 Aug 2016  Raanan Chermoni & Jaroslaw Wroblewski  
3  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  8 Oct 2015  Raanan Chermoni & Jaroslaw Wroblewski  
4  39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  29  8 Jan 2015  Raanan Chermoni & Jaroslaw Wroblewski  
5  29 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  23      
More Prime 21tuplets 
A prime ktuplet 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 b_{1} < b_{2} < ... < b_{k}, b_{k} − b_{1} = s and, for every prime q, not all the residues modulo q are represented by b_{1}, b_{2}, ..., b_{k}. A prime ktuplet is then defined as a sequence of consecutive primes {p_{1}, p_{2}, ..., p_{k}} such that for every prime q, not all the residues modulo q are represented by p_{1}, p_{2}, ..., p_{k}, p_{k} − p_{1} = 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 5tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.
Patterns of Prime ktuplets
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 Online 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, a^{N} = 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 (APRTCLE 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 FASTECPP, 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/p^{2}) = ∑_{n = 1 to ∞} 1/n^{2} = π^{2}/6.
But π^{2} 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 A √x 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 C_{2} x / (log x)^{2},
where C_{2} = ∏_{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 JingRun 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 HardyLittlewood Prime ktuple Conjecture
The Partitio Numerorum: III paper [HL22] goes on to formulate a general conjecture concerning the distribution of arbitrary groups of prime numbers (The ktuplets of this site are special cases): Let b_{1}, b_{2}, ..., b_{k} be k distinct integers. Then the number of groups of primes N + b_{1}, N + b_{2}, ..., N + b_{k} between 2 and x is approximately
H_{k} C_{k} ∫_{u = 2 to x} du / (log u)^{k},
where
H_{k} = ∏_{p prime, p ≤ k} p^{k − 1} (p − v) / (p − 1)^{k} ∏_{p prime, p > k, pD} (p − v) / (p − k),
C_{k} = ∏_{p prime, p > k} p^{k − 1} (p − k) / (p − 1)^{k},
v = v(p) is the number of distinct remainders of b_{1}, b_{2}, ..., b_{k} modulo p and D is the product of the differences b_{i} − b_{j}, 1 ≤ i < j ≤ k.
The first product in H_{k} is over the primes not greater than k, the second is over the primes greater than k which divide D and the product C_{k} is over all primes greater than k. If you put k = 2, b_{1} = 0 and b_{2} = 2, then v(2) = 1, v(p) = p − 1 for p > 2, H_{2} = 2, and C_{k} = C_{2}, the twin prime constant given above.
It is worth pointing out that with modern mathematical software the prime ktuplet constants C_{k} 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 C_{k} = − ∑_{n = 2 to ∞} log [ζ(n) ∏_{p prime, p ≤ k} (1 − 1/p^{n})] / n ∑_{dn} μ(n/d) (k^{d} − k).
[BLS75] John Brillhart, D.H. Lehmer & J.L. Selfridge, New primality criteria and factorizations of 2^{m} ± 1, Math. Comp. 29 (1975), 620647.
[CD93] C.K.Caldwell & H. Dubner, Primorial, factorial and multifactorial primes, Math. Spectrum 26 (1993/94), 17.
[F97f] Tony Forbes, Prime 17tuplet, NMBRTHRY Mailing List, September 1997.
[F02] Tony Forbes, Titanic prime quintuplets, M500 189 (December, 2002), 1213.
[F09] Tony Forbes, Gigantic prime triplets, M500 226 (February, 2009), 1819.
[Guy94] Richard K. Guy, Unsolved Problems in Number Theory, second edn., SpringerVerlag, 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), 170.
[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., SpringerVerlag, 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), 45.
[R97a] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1997.
[R97b] Warut Roonguthai, Large prime quadruplets, M500 158 (November 1997), 15.