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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.
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.
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.
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.
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 prime quadruplet of 2000 digits, found by Gerd Lamprecht in Oct 2017:
10^{1999} + 205076414983951,
10^{1999} + 205076414983953,
10^{1999} + 205076414983957,
10^{1999} + 205076414983959.
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.
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.
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@pzktupel.de.
2996863034895 * 2^{1290000} ± 1  Tom Greer, TwinGen, PrimeGrid, LLR  
3756801695685 * 2^{666669} ± 1  Timothy Winslow, TwinGen, PrimeGrid, LLR  
65516468355 * 2^{333333} ± 1  Peter Kaiser, NewPGen, PrimeGrid, TPS, LLR  
160204065 * 2^{262148} ± 1  Erwin Doescher, LLR  
12770275971 * 2^{222225} ± 1  Bo Tornberg, TwinGen, LLR  
70965694293 * 2^{200006} ± 1  S. Urushihata  
66444866235 * 2^{200003} ± 1  S. Urushihata  
4884940623 * 2^{198800} ± 1  Michael Kwok, PSieve, LLR  
2003663613 * 2^{195000} ± 1  Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Michael Kwok, Andrea Pacini & Rytis Slatkevicius 

17976255129 * 2^{183241} ± 1  Frank Doornink, TwinGen, OpenPFGW 
more 
4111286921397 * 2^{66420}  1 + d, d = 0, 2, 6  Peter Kaiser, Polysieve, LLR, Primo  
6521953289619 * 2^{55555}  5 + d, d = 0, 4, 6  Peter Kaiser  
4207993863 * 2^{38624 }  1 + d, d = 0, 2, 6  Frank Doornink, NewPGen, LLR, Primo  
14059969053 * 2^{36672}  5 + d, d = 0, 4, 6  Serge Batalov, NewPgen, OpenPFGW, Primo  
3221449497221499 * 2^{34567}  1 + d, d = 0, 2, 6  Peter Kaiser, NewPGen, LLR, OpenPFGW  
1288726869465789 * 2^{34567}  5 + d, d = 0, 4, 6  Peter Kaiser  
647935598824239 * 2^{33619}  1 + d, d = 0, 2, 6  Peter Kaiser, Primo  
209102639346537 * 2^{33620}  1 + d, d = 0, 2, 6  Peter Kaiser, Primo  
185353103135997 * 2^{33620}  1 + d, d = 0, 2, 6  Peter Kaiser, Primo  
162615027598677 * 2^{33620}  1 + d, d = 0, 2, 6  Peter Kaiser, Primo 
more 
667674063382677 * 2^{33608}  1 + d, d = 0, 2, 6, 8  Peter Kaiser, Primo  
4122429552750669 * 2^{16567}  1 + d, d = 0, 2, 6, 8  Peter Kaiser, GSIEVE, NewPGen, LLR, Primo  
101406820312263 * 2^{12042}  1 + d, d = 0, 2, 6, 8  Serge Batalov, OpenPFGW, NewPGen, Primo  
2673092556681 * 15^{3048}  4 + d, d = 0, 2, 6, 8  Serge Batalov, OpenPFGW, NewPGen, Primo  
2339662057597 * 10^{3490} + 1 + d, d = 0, 2, 6, 8  Serge Batalov, OpenPFGW, NewPGen, Primo  
305136484659 * 2^{11399}  1 + d, d = 0, 2, 6, 8  Serge Batalov, OpenPFGW, NewPGen, Primo  
722047383902589 * 2^{11111}  1 + d, d = 0, 2, 6, 8  Reto Keiser, NewPGen, PFGW, Primo  
43697976428649 * 2^{9999}  1 + d, d = 0, 2, 6, 8  Peter Kaiser  
46359065729523 * 2^{8258}  1 + d, d = 0, 2, 6, 8  Reto Keiser, NewPGen, PFGW, Primo  
1367848532291 * 5591# / 35  1 + d, d = 0, 2, 6, 8  Norman Luhn, NewPGen, PFGW, Primo 
more 
566761969187 * 4733# / 2  8 + d, d = 0, 4, 6, 10, 12  Serge Batalov, NewPGen, OpenPFGW, Primo  
126831252923413 * 4657# / 273 + 1 + d, d = 0, 2, 6, 8, 12  Peter Kaiser, Primo  
394254311495 * 3733# / 2  8 + d, d = 0, 4, 6, 10, 12  Serge Batalov, NewPGen, OpenPFGW, Primo  
2316765173284 * 3600# + 16061 + d, d = 0, 2, 6, 8, 12  Norman Luhn, Primo  
163252711105 * 3371# / 2  8 + d, d = 0, 4, 6, 10, 12  Serge Batalov, OpenPFGW, NewPGen, Primo  
9039840848561 * 3299# / 35  5 + d, d = 0, 4, 6, 10, 12  Serge Batalov, OpenPFGW, NewPGen, Primo  
699549860111847 * 2^{4244}  1 + d, d = 0, 2, 6, 8, 12  Reto Keiser, R. Gerbicz, PFGW, Primo  
405095429109490796 * 2683# + 16057 + d, d = 0, 4, 6, 10, 12  Michael Bell, Rieminer, ECPPDJ  
566650659276 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12  David Broadhurst, Primo, OpenPFGW  
554729409262 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12  David Broadhurst, Primo, OpenPFGW 
more 
2^{3700} + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16  Vidar Nakling, Primo, Sixfinder ( based on Riecoin miners ) 

28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16  Norman Luhn, APSIEVE, Primo  
6646873760397777881866826327962099685830865900246688640856 * 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16  Vidar Nakling, Primo  
29720510172503062360713760607985203309940766118866743491802189150471978534404249 * 2^{2299} + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16 
Riecoin #822096  
29749903422302373222996698880833194129159047179535887991184960156219652236318921 * 2^{2293} + 679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16 
Riecoin #793872  
29696802688480280387313212926526693549449146292085717645262228449092881114972806 * 2^{2290} + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16 
Riecoin #838224  
29744205023784420961031622414734790416939049568996819659808238403983863222665068 * 2^{2288} + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16 
Riecoin #834192  
29707412718946949415029080194980493978605678414396606766712262274235284928962561 * 2^{2278} + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16 
Riecoin #814032  
29696978890366869883141509418765838581871522982358338407613039711378021084519043 * 2^{2259} + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16 
Riecoin #805968  
29691575669072177222494655186416928710256802541243921484227880404600991044790342 * 2^{2259} + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16 
Riecoin #797904 
more 
113225039190926127209 * 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20  Peter Kaiser  
3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473 9224962020153653925994202321897236902676229040360901005487309186655777663859063397693729163631275766 0779987530903845763711693853827939526026506444774774261236889041020217108597484837589978261046949778 7199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089 * 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 
Vidar Nakling, Rieminer 0.9, Primo  
115828580393941 * 1200# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20  Norman Luhn, Primo  
4733578067069 * 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20  Norman Luhn, Primo  
687001431518312990252195799540952 * 719# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20  Michalis Christou, Rieminer  
686636073174158279347746711902518 * 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20  Michalis Christou, Rieminer  
686488342697495738978150794512038 * 719# + 1277156391416021 + d, d = 0, 2, 6, 8, 12, 18, 20  Michalis Christou, Rieminer  
686305940768787196771563962517233 * 719# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20  Michalis Christou, Rieminer  
686129792256610907998640667932122 * 719# + 701889794782061 + d, d = 0, 2, 6, 8, 12, 18, 20  Michalis Christou, Rieminer  
685817639451814894948541841801955 * 719# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20  Michalis Christou, Rieminer 
more 
6879356578124627875380298699944709053335 * 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  Michalis Christou  
54598824190010361875282469578684418459657573362461324471660422883073099662240278837985413217294784653805 * 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
Riecoin #1607166  
237290937625019988409934680338216405908629349352492341129431599973490073614754863588338476036934867547671407908 * 487# + 1146773 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
Riecoin #1600958  
188273324392097141944873869557423547058811920840483304365112457383885407879644413861445197917160744 * 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
Riecoin #1600978  
6068138408292784654794269848877333341123929067736255007020032491702134706361073607222476583743922495929518535 * 487# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 
Riecoin #1600993  
6492845263546348546 * 700# + 226449521 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  Thomas Nguyen, Rieminer 0.91  
652860139668148506027538015230153318100639856923430236186190699612250481487112024837265410828490443427344306052 * 467# + 1418575498583 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
Riecoin #1604890  
1003196298617375646416642308462802536422799049663270195408652126607761421584998613928566265881964993799 * 491# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 
Riecoin #1604885  
6906530913807107618746553985178874306453389164948020895483633515626362605252139055492044931282748833678120293 * 467# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 
Riecoin #1567265  
359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26  Norman Luhn, VFYPR 
more 
3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501 * 463# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
Riecoin #1567399  
1620259924615470570706663156278905026372754732844252658390408090245313172792664271166384219300680488342402961778 * 450# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
Riecoin #1566093  
387833514641724600357029749119397331285062620621983133723181869572568059514167753188325960698719230 * 467# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
Riecoin #1605403  
352360483181346865458241271679723612698522403422825810225007396594617602134749714450854301781722077875166 * 457# + 302000014586509 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
Riecoin #1640522  
40893595297845006551741048717748959451570266851095389722761855002653709793065456232477944049520841797242 * 457# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
Riecoin #1576463  
1285897436414229397879253137835709328420499854296690250107786136168118236699867925624486888053153988984555053 * 443# + 27899359258009 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
Riecoin #1636021  
1574649584907747555706133630785765300471148459969743147116987603964781873723920925748330533139472006902 * 457# + 145799289681161 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
Riecoin #1628519  
10850212521100776545123088182949734846433679305043970388545713677317869416103316562935316943942840819 * 457# + 1833994713165731 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
Riecoin #1646284  
732510298464897302406863094406522970022698347078480858977704780162529246459062655850703116903119380663871601 * 440# + 114521428533971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 
Riecoin #1558658  
1657818470571599561261598554164065452026126127696138156496865986299371039919391471562919572623342964479 * 449# + 226554621544609 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 
Riecoin #1632244 
more 
14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550 * 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 
Riecoin #1579367  
290901656335108169864195656135043662615199446375386143995339722400236057821426952579658098504166333411889 * 401# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
Riecoin #1551825  
33521646378383216495527 * 331# + 4700094892301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  Thomas Nguyen, Rieminer 0.91, MPZAPRCL 

772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32  Norman Luhn  
7425 * 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  Roger Thompson  
118557188915212 * 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32  Norman Luhn  
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 53586844409797545 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo  
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 51143234991402697 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo  
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 50679161987995696 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo  
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49561325184911775 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 
more 
13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 46622982649030457 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 30796489110940369 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20731977215353082 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20118509988610513 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 15866045335517629 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 5238271627884665 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 1296173254392493 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 
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13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 
Peter Kaiser, David Stevens, Polysieve, PFGW, Primo 

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

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

9985637467 * 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  Roger Thompson 

9985397181 * 139# + 249386599747880711 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  Roger Thompson 

59125383480754 * 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42  Michael Stocker 

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

450725899 * 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  Martin Raab 

14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  Roger Thompson 

8486221 * 107# + 4549290807806861 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42  Dirk Augustin & Jens Kruse Andersen 
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9985637467 * 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  Roger Thompson  
4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48  Norman Luhn 

14815550 * 107# + 4385574275277313 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48  Roger Thompson 

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

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

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

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

1000000000000000027545153594708289884461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  Norman Luhn 

1000000000000000014210159036148101380473 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48  Norman Luhn 

1000000000000000014210159036148101380471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48  Norman Luhn 
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14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  Roger Thompson  
381955327397348 * 80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50  Norman Luhn 

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

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

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

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

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

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

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

101885197790002105359911556070661 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50  Jens Kruse Andersen 
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33554294028531569 * 61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  Norman Luhn  
322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  Roger Thompson  
10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  Roger Thompson  
94 * 79# + 1341680294611244014367 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  Roger Thompson  
3684 * 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56  Roger Thompson  
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  Jens Kruse Andersen  
99999999948164978600250563546411 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  Jörg Waldvogel and Peter Leikauf  
1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  Hans Rosenthal & Jens Kruse Andersen  
1100916249233879857334075234831 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56  Hans Rosenthal & Jens Kruse Andersen  
1003234871202624616703163933857 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56  Roger Thompson 
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322255 * 73# + 1354238543317302647 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  Roger Thompson  
94 * 79# + 1341680294611244014363 + d, d= 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  Roger Thompson  
3684 * 73# + 880858118723497737827 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  Roger Thompson  
1003234871202624616703163933853 + d, d= 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  Roger Thompson  
11413975438568556104209245223 + d, d= 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60  Roger Thompson  
5867208169546174917450988007 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659017 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  Raanan Chermoni & Jaroslaw Wroblewski  
4668263977931056970475231227 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  Raanan Chermoni & Jaroslaw Wroblewski  
4652363394518920290108071177 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  Raanan Chermoni & Jaroslaw Wroblewski  
4483200447126419500533043997 + d, d= 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60  Raanan Chermoni & Jaroslaw Wroblewski 
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3684 * 73# + 880858118723497737821 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  Roger Thompson  
100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66  Roger Thompson  
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66  Roger Thompson  
11410793439953412180643704677 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66  Roger Thompson  
5867208169546174917450988001 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  Raanan Chermoni & Jaroslaw Wroblewski  
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659011 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  Raanan Chermoni & Jaroslaw Wroblewski  
4668263977931056970475231221 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66  Raanan Chermoni & Jaroslaw Wroblewski  
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66  Raanan Chermoni & Jaroslaw Wroblewski 
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5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  Raanan Chermoni & Jaroslaw Wroblewski  
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70  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  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  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  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  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  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  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  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  Raanan Chermoni & Jaroslaw Wroblewski 
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622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76  Raanan Chermoni & Jaroslaw Wroblewski  
248283957683772055928836513597 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76  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  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  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  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  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  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  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  Raanan Chermoni & Jaroslaw Wroblewski  
{37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} 
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1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
1188350591359110800209379560799 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
1153897621507935436463788957529 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
1135540756371356698957890225821 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
1126002593922465663847897293731 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
1094372814043722195189448411199 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
1060475118776959297139870952701 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
999627565307688186459783232931 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
957278727962618711849051282459 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski  
839013472011818416634745523991 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80  Raanan Chermoni & Jaroslaw Wroblewski 
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622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84  Raanan Chermoni & Jaroslaw Wroblewski  
248283957683772055928836513589 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  Raanan Chermoni & Jaroslaw Wroblewski  
138433730977092118055599751669 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  Raanan Chermoni & Jaroslaw Wroblewski  
39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84  Raanan Chermoni & Jaroslaw Wroblewski  
{29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} 
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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.
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.
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.
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.
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 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.
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[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.
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