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]. 
The Largest Known Twin Primes  Digits  When  Additions 
2996863034895 • 2^{1290000} ± 1  388342  19 Sep 2016  Tom Greer, TwinGen, PrimeGrid, LLR 
3756801695685 • 2^{666669} ± 1  200700  26 Dec 2011  Timothy Winslow, TwinGen, PrimeGrid, LLR 
65516468355 • 2^{333333} ± 1  100355  15 Aug 2009  Peter Kaiser, NewPGen, PrimeGrid, TPS, LLR 
160204065 • 2^{262148} ± 1  78923  8 Jul 2021  Erwin Doescher, LLR 
12770275971 • 2^{222225} ± 1  66907  4 Jul 2017  Bo Tornberg, TwinGen, LLR 
12599682117 • 2^{211088} ± 1  63554  22 Feb 2022  Michael Kwok, PSieve, LLR 
12566577633 • 2^{211088} ± 1  63554  22 Feb 2022  Michael Kwok, PSieve, LLR 
70965694293 • 2^{200006} ± 1  60219  2 Apr 2016  S. Urushihata 
66444866235 • 2^{200003} ± 1  60218  2 Apr 2016  S. Urushihata 
4884940623 • 2^{198800} ± 1  59855  3 Jul 2015  Michael Kwok, PSieve, LLR 
More Twin Primes 
The Largest Known Prime Triplets  Digits  When  Additions  Certificates 
4111286921397 • 2^{66420} − 1 + d, d = 0, 2, 6  20008  24 Apr 2019  Peter Kaiser, Polysieve, LLR, Primo  click 
6521953289619 • 2^{55555} − 5 + d, d = 0, 4, 6  16737  30 Apr 2013  Peter Kaiser  click 
56667641271 • 2^{44441} − 1 + d, d = 0, 2, 6  13389  1 Apr 2022  Stephan Schöler, NewPGen, OpenPFGW; Oliver Kruse, Primo  click 
4207993863 • 2^{38624 } − 1 + d, d = 0, 2, 6  11637  5 Jun 2021  Frank Doornink, NewPGen, LLR, Primo  click 
14059969053 • 2^{36672} − 5 + d, d = 0, 4, 6  11050  17 Jun 2018  Serge Batalov, NewPGen, OpenPFGW, Primo  click 
3221449497221499 • 2^{34567} − 1 + d, d = 0, 2, 6  10422  2 Sep 2015  Peter Kaiser, NewPGen, LLR, OpenPFGW, Primo  click 
1288726869465789 • 2^{34567} − 5 + d, d = 0, 4, 6  10421  23 Apr 2014  Peter Kaiser, Primo  click 
647935598824239 • 2^{33619} − 1 + d, d = 0, 2, 6  10136  22 May 2019  Peter Kaiser, Primo  click 
209102639346537 • 2^{33620} − 1 + d, d = 0, 2, 6  10135  22 May 2019  Peter Kaiser, Primo  click 
185353103135997 • 2^{33620} − 1 + d, d = 0, 2, 6  10135  22 May 2019  Peter Kaiser, Primo  click 
More Prime Triplets 
The Largest Known Prime Quadruplets  Digits  When  Additions  Certificates 
667674063382677 • 2^{33608} − 1 + d, d = 0, 2, 6, 8  10132  27 Feb 2019  Peter Kaiser, Primo  click 
4122429552750669 • 2^{16567} − 1 + d, d = 0, 2, 6, 8  5003  10 Mar 2016  Peter Kaiser, GSIEVE, NewPGen, LLR, Primo  click 
101406820312263 • 2^{12042} − 1 + d, d = 0, 2, 6, 8  3640  13 Jun 2018  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
2673092556681 • 15^{3048} − 4 + d, d = 0, 2, 6, 8  3598  14 Sep 2015  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
2339662057597 • 10^{3490} + 1 + d, d = 0, 2, 6, 8  3503  21 Dec 2013  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
305136484659 • 2^{11399} − 1 + d, d = 0, 2, 6, 8  3443  28 Sep 2013  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
722047383902589 • 2^{11111} − 1 + d, d = 0, 2, 6, 8  3360  20 Apr 2013  Reto Keiser, NewPGen, OpenPFGW, Primo  click 
585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8  3344  06 Mar 2022  Peter Kaiser, OpenPFGW, Primo  click 
43697976428649 • 2^{9999} − 1 + d, d = 0, 2, 6, 8  3024  24 Mar 2012  Peter Kaiser, Primo 3.0.9  click 
10^{2999} + 339930644528851 + d, d = 0, 2, 6, 8  3000  04 May 2022  Norman Luhn, NewPGen, OpenPFGW, Primo  click 
More Prime Quadruplets 
The Largest Known Prime Quintuplets  Digits  When  Additions  Certificates 
585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8, 12  3344  06 Mar 2022  Peter Kaiser, OpenPFGW, Primo  click 
566761969187 • 4733# / 2 − 8 + d, d = 0, 4, 6, 10, 12  2034  06 Dec 2020  Serge Batalov, NewPGen, OpenPFGW, Primo  click 
126831252923413 • 4657# / 273 + 1 + d, d = 0, 2, 6, 8, 12  2002  8 Nov 2020  Peter Kaiser, Primo  click 
394254311495 • 3733# / 2 − 8 + d, d = 0, 4, 6, 10, 12  1606  Nov 2017  Serge Batalov, NewPGen, OpenPFGW, Primo  click 
2316765173284 • 3600# + 16061 + d, d = 0, 2, 6, 8, 12  1543  16 Oct 2016  Norman Luhn, Primo  click 
163252711105 • 3371# / 2 − 8 + d, d = 0, 4, 6, 10, 12  1443  1 Jan 2014  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
9039840848561 • 3299# / 35 − 5 + d, d = 0, 4, 6, 10, 12  1401  Dec 2013  Serge Batalov, OpenPFGW, NewPGen, Primo  click 
699549860111847 • 2^{4244} − 1 + d, d = 0, 2, 6, 8, 12  1293  3 Dec 2013  Reto Keiser, R. Gerbicz, OpenPFGW, Primo  click 
10^{1199} + 20483870459152351 + d, d = 0, 2, 6, 8, 12  1200  03 Mar 2023  Norman Luhn, OpenPFGW, Primo 3.0.9  click 
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 
The Largest Known Prime Sextuplets  Digits  When  Additions  Certificates 
2^{3700} + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16  1114  8 Nov 2021  Vidar Nakling, Primo, Sixfinder ( based on Riecoin miners ) 
click 
28993093368077 • 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16  1037  14 Mar 2016  Norman Luhn, APSIEVE, Primo  click 
6646873760397777881866826327962099685830865900246688640856 • 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16  780  8 Nov 2018  Vidar Nakling, Primo   
29720510172503062360713760607985203309940766118866743491802189150471978534404249 • 2^{2299} + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16 
772  28 Jan 2018  Riecoin #822096   
29749903422302373222996698880833194129159047179535887991184960156219652236318921 • 2^{2293} + 679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16 
770  28 Sep 2017  Riecoin #793872   
29696802688480280387313212926526693549449146292085717645262228449092881114972806 • 2^{2290} + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16 
769  25 Feb 2018  Riecoin #838224   
29744205023784420961031622414734790416939049568996819659808238403983863222665068 • 2^{2288} + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16 
769  18 Feb 2018  Riecoin #834192   
29707412718946949415029080194980493978605678414396606766712262274235284928962561 • 2^{2278} + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16 
766  14 Jan 2018  Riecoin #814032   
29696978890366869883141509418765838581871522982358338407613039711378021084519043 • 2^{2259} + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16 
766  31 Dec 2017  Riecoin #805968   
29691575669072177222494655186416928710256802541243921484227880404600991044790342 • 2^{2259} + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16 
760  16 Dec 2017  Riecoin #797904   
More Prime Sextuplets 
The Largest Known Prime Septuplets  Digits  When  Additions  Certificates 
113225039190926127209 • 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20  1002  27 Jan 2021  Peter Kaiser  click 
3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473 9224962020153653925994202321897236902676229040360901005487309186655777663859063397693729163631275766 0779987530903845763711693853827939526026506444774774261236889041020217108597484837589978261046949778 7199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089 • 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 
527  16 Jun 2019  Vidar Nakling,
rieMiner 0.9, Primo 
 
115828580393941 • 1200# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20  515  18 Jan 2018  Norman Luhn, Primo   
4733578067069 • 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20  402  09 May 2016  Norman Luhn, Primo   
17823436097302039011383530397574638170734 • 771# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20  362  12 Jan 2023  Michalis Christou, rieMiner   
17823429667990098910404937524728999750239 • 771# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20  362  12 Jan 2023  Michalis Christou, rieMiner   
17823427931355237791135579059273253342104 • 771# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20  362  12 Jan 2023  Michalis Christou, rieMiner   
17823424548404400585693182916303260773687 • 771# + 145844141558231 + d, d = 0, 2, 6, 8, 12, 18, 20  362  12 Jan 2023  Michalis Christou, rieMiner   
17823423197963241682981949738836473771455 • 771# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20  362  12 Jan 2023  Michalis Christou, rieMiner   
17823416716191253381804284202848071984717 • 771# + 145844141558231 + d, d = 0, 2, 6, 8, 12, 18, 20  362  12 Jan 2023  Michalis Christou, rieMiner   
More Prime Septuplets 
The Largest Known Prime Octuplets  Digits  When  Additions  
17823192282008874449172703428792123231110 • 771# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  362  12 Jan 2023  Michalis Christou, rieMiner  
531258360785860208657753 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  333  30 Sep 2022  Peter Kaiser, Primo  
530956818040688210255681 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  333  30 Sep 2022  Peter Kaiser, Primo  
697723422149271424870176724491962624555 • 701# + 145888993435301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  332  28 Apr 2022  Michalis Christou, rieMiner 0.91  
5586218959960365309179 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  331  12 Sep 2022  Peter Kaiser, Primo  
4869586684665128135306 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  331  30 Sep 2022  Peter Kaiser, Primo  
6879356578124627875380298699944709053335 • 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  324  12 Mar 2021  Michalis Christou, rieMiner 0.91  
54598824190010361875282469578684418459657573362461324471660422883073099662240278837985413217294784653805 • 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
316  30 Oct 2021  Riecoin #1607166  
237290937625019988409934680338216405908629349352492341129431599973490073614754863588338476036934867547671407908 • 487# + 1146773 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
312  20 Oct 2021  Riecoin #1600958  
188273324392097141944873869557423547058811920840483304365112457383885407879644413861445197917160744 • 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
310  20 Oct 2021  Riecoin #1600978  
6068138408292784654794269848877333341123929067736255007020032491702134706361073607222476583743922495929518535 • 487# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 
310  20 Oct 2021  Riecoin #1600993  
More Prime Octuplets  
Prime Octuplets, found by Riecoin 
The Largest Known Prime Nonuplets  Digits  When  Additions  
7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341 • 503# + 220469307413891 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
309  16 Feb 2022  Bielawski Mathematicians  
3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501 • 463# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
302  22 Aug 2021  Riecoin #1567399  
1628310212800330650680875193463302528297524858575252703732019584033146050060402902119214625070 • 491# + 437171043550511 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
297  25 Mar 2023  Riecoin #1902419  
1620259924615470570706663156278905026372754732844252658390408090245313172792664271166384219300680488342402961778 • 450# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
296  20 Aug 2021  Riecoin #1566093  
387833514641724600357029749119397331285062620621983133723181869572568059514167753188325960698719230 • 467# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
294  27 Oct 2021  Riecoin #1605403  
352360483181346865458241271679723612698522403422825810225007396594617602134749714450854301781722077875166 • 457# + 302000014586509 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
292  27 Dec 2021  Riecoin #1640522  
40893595297845006551741048717748959451570266851095389722761855002653709793065456232477944049520841797242 • 457# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
291  7 Sep 2021  Riecoin #1576463  
1285897436414229397879253137835709328420499854296690250107786136168118236699867925624486888053153988984555053 • 443# + 27899359258009 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
291  19 Dec 2021  Riecoin #1636021  
425637736526956247129170414648410773829792758837126862506126866133457379262600484880404875012755358905043168102 • 439# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
291  14 Mar 2022  Riecoin #1685127  
1574649584907747555706133630785765300471148459969743147116987603964781873723920925748330533139472006902 • 457# + 145799289681161 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
290  6 Dec 2021  Riecoin #1628519  
More Prime Nonuplets  
Prime Nonuplets, found by Riecoin 
The Largest Known Prime 10tuplets  Digits  When  Additions  
14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550 • 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 
282  12 Sep 2021  Riecoin #1579367  
290901656335108169864195656135043662615199446375386143995339722400236057821426952579658098504166333411889 • 401# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
269  27 Jul 2021  Riecoin #1551825  
14257429881902877844339877915045298096140599288873476083093543949692946630381247693511330479634493 • 367# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
246  18 Aug 2022  Riecoin #1775788  
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 

772556746441918 • 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32  136  9 Feb 2017  Norman Luhn  
7425 • 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  120  27 May 2016  Roger Thompson  
118557188915212 • 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  118  27 Jun 2014  Norman Luhn  
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 53586844409797545 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
108  23 Sep 2019  Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo 

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

1000000000000000000000000000000002760339313453283246757 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42  55  04 Oct 2022  Norman Luhn 

More Prime 12tuplets 
The Largest Known Prime 13tuplets  Digits  When  Additions  
9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  66  1 Oct 2021  Roger Thompson  
4135997219394611 • 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48  61  23 Mar 2017  Norman Luhn 

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

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

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

381955327397348 • 80# + 18393211 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48  46  28 Dec 2007  Norman Luhn 

381955327397348 • 80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48  46  28 Dec 2007  Norman Luhn 

100000000000000000000006149198224095343810309 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48  45  02 Mar 2022  Norman Luhn 

100000000000000000000004356680452416578030761 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  45  05 Feb 2022  Norman Luhn 

100000000000000000000002004740564798426955633 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48  45  06 Mar 2022  Norman Luhn 

More Prime 13tuplets 
The Largest Known Prime 14tuplets  Digits  When  Additions  
14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  50  5 Feb 2013  Roger Thompson  
381955327397348 • 80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50  46  28 Dec 2007  Norman Luhn 

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

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

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

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

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

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

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

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

More Prime 14tuplets 
The Largest Known Prime 15tuplets  Digits  When  Additions  
33554294028531569 • 61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  40  25 Jan 2017  Norman Luhn  
322255 • 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  35  18 Nov 2016  Roger Thompson  
10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  35  4 Sep 2012  Roger Thompson  
94 • 79# + 1341680294611244014367 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  33  5 Feb 2021  Roger Thompson  
3684 • 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  33  5 Feb 2021  Roger Thompson  
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  33  14 Apr 2008  Jens Kruse Andersen  
99999999948164978600250563546411 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  32  29 Nov 2004  Jörg Waldvogel and Peter Leikauf  
1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  31  16 Oct 2003  Hans Rosenthal & Jens Kruse Andersen  
1100916249233879857334075234831 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  31  16 Oct 2003  Hans Rosenthal & Jens Kruse Andersen  
1003234871202624616703163933857 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  31  9 Aug 2012  Roger Thompson  
More Prime 15tuplets 
The Largest Known Prime 16tuplets  Digits  When  Additions  
322255 • 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  35  18 Nov 2016  Roger Thompson  
94 • 79# + 1341680294611244014363 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  33  5 Feb 2021  Roger Thompson  
3684 • 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  33  5 Feb 2021  Roger Thompson  
1003234871202624616703163933853 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  31  9 Aug 2012  Roger Thompson  
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  29  2 Jan 2012  Roger Thompson  
5867208169546174917450988007 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659017 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4668263977931056970475231227 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4652363394518920290108071177 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4483200447126419500533043997 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 16tuplets 
The Largest Known Prime 17tuplets  Digits  When  Additions  
3684 • 73# + 880858118723497737821 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  33  5 Feb 2021  Roger Thompson  
100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66  30  19 Feb 2013  Roger Thompson  
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66  29  2 Jan 2012  Roger Thompson  
11410793439953412180643704677 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66  29  2 Jan 2012  Roger Thompson  
5867208169546174917450988001 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659011 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4668263977931056970475231221 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 17tuplets 
The Largest Known Prime 18tuplets  Digits  When  Additions  
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  11 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Mar 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4652363394518920290108071167 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
4483200447126419500533043987 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  4 Jan 2014  Raanan Chermoni & Jaroslaw Wroblewski  
3361885098594416802447362317 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  30 Jul 2013  Raanan Chermoni & Jaroslaw Wroblewski  
3261917553005305074228431077 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  30 Jul 2013  Raanan Chermoni & Jaroslaw Wroblewski  
3176488693054534709318830357 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  30 Jul 2013  Raanan Chermoni & Jaroslaw Wroblewski  
2650778861583720495199114537 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  25 Feb 2013  Raanan Chermoni & Jaroslaw Wroblewski  
2406179998282157386567481197 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  28  31 Dec 2012  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 18tuplets 
The Largest Known Prime 19tuplets  Digits  When  Additions  
622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76  30  27 Dec 2018  Raanan Chermoni & Jaroslaw Wroblewski  
248283957683772055928836513597 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  30  1 Aug 2016  Raanan Chermoni & Jaroslaw Wroblewski  
138433730977092118055599751677 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  30  8 Oct 2015  Raanan Chermoni & Jaroslaw Wroblewski  
39433867730216371575457664407 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  29  8 Jan 2015  Raanan Chermoni & Jaroslaw Wroblewski  
2406179998282157386567481191 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  28  31 Dec 2012  Raanan Chermoni & Jaroslaw Wroblewski  
2348190884512663974906615481 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  28  17 Dec 2012  Raanan Chermoni & Jaroslaw Wroblewski  
917810189564189435979968491 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  27  29 May 2011  Raanan Chermoni & Jaroslaw Wroblewski  
656632460108426841186109951 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  27  19 Feb 2011  Raanan Chermoni & Jaroslaw Wroblewski  
630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76  27  9 Feb 2011  Raanan Chermoni & Jaroslaw Wroblewski  
{37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}  
More Prime 19tuplets 
The Largest Known Prime 20tuplets  Digits  When  Additions  
1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  23 May 2021  Raanan Chermoni & Jaroslaw Wroblewski  
1188350591359110800209379560799 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  21 Jan 2021  Raanan Chermoni & Jaroslaw Wroblewski  
1153897621507935436463788957529 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  26 Dec 2020  Raanan Chermoni & Jaroslaw Wroblewski  
1135540756371356698957890225821 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  31  19 Dec 2020  Raanan Chermoni & Jaroslaw Wroblewski  
1126002593922465663847897293731 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  31  17 Nov 2020  Raanan Chermoni & Jaroslaw Wroblewski  
1094372814043722195189448411199 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  31  20 Oct 2020  Raanan Chermoni & Jaroslaw Wroblewski  
1060475118776959297139870952701 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  31  18 Sep 2020  Raanan Chermoni & Jaroslaw Wroblewski  
999627565307688186459783232931 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  30  19 Jun 2020  Raanan Chermoni & Jaroslaw Wroblewski  
957278727962618711849051282459 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  30  23 Mar 2020  Raanan Chermoni & Jaroslaw Wroblewski  
839013472011818416634745523991 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  30  28 Oct 2020  Raanan Chermoni & Jaroslaw Wroblewski  
More Prime 20tuplets 
The Largest Known Prime 21tuplets  Digits  When  Additions  
622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84  30  27 Dec 2018  Raanan Chermoni & Jaroslaw Wroblewski  
248283957683772055928836513589 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  30  1 Aug 2016  Raanan Chermoni & Jaroslaw Wroblewski  
138433730977092118055599751669 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  30  8 Oct 2015  Raanan Chermoni & Jaroslaw Wroblewski  
39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  29  8 Jan 2015  Raanan Chermoni & Jaroslaw Wroblewski  
{29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}        
More 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.