At this site I have collected together all the largest known examples of certain types of dense clusters of prime numbers. The idea is to generalise the notion of prime twins - pairs of prime numbers {p, p + 2} - to groups of three or more.
Prepared by Tony Forbes; anthony.d.forbes@gmail.com.
Site address: http://anthony.d.forbes.googlepages.com/ktuplets.htm.
25 October 2017
Prime quadruplet (SMALLEST WITH 2000 DIGITS)
10^1999 + 205076414983951 + d, d = 0, 2, 6, 8 (2000 digits, 25 Oct 2017, Gerd Lamprecht, PRIMO)
4 July 2017
Prime octuplet (NEW WORLD RECORD!)
359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
Prime septuplets
771620215080738 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (305 digits, 4 Jul 2017, Norman Luhn, VFYPR)
375681809246516 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (305 digits, 4 Jul 2017, Norman Luhn, VFYPR)
359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
334063689033226 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
295031928451848 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
263868283973329 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
164124046378678 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
126443148358786 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
24 May 2017
Prime 20-tuplets (including NEW WORLD RECORD!)
352259532126245901475150592651 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, 24 May 2017, Raanan Chermoni & Jaroslaw Wroblewski)
349021296319127268299400177269 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, 24 May 2017, Raanan Chermoni & Jaroslaw Wroblewski)
320430661688896578454772807699 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, 24 May 2017, Raanan Chermoni & Jaroslaw Wroblewski)
23 March 2017
Prime 13-tuplet (NEW WORLD RECORD! And with the 'forgotten' pattern, {0, 2, 12, ...})
4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (61 digits, 23 Mar 2017, Norman Luhn)
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, inter-decal prime quartet (!) and prime decade - a reference to the pattern made by their decimal digits. All primes greater than 5 end in one of 1, 3, 7 or 9, and the four primes in a (large) quadruplet always occur in the same ten-block. Hence there must be exactly one with each of these unit digits. And just to illustrate the point, here is another example; the smallest prime quadruplet of 50 digits, found by G. John Stevens in 1995 [S95]:
10000000000000000000000000000000000000000058537891,
10000000000000000000000000000000000000000058537893,
10000000000000000000000000000000000000000058537897,
10000000000000000000000000000000000000000058537899.
We can go on to define prime quintuplets, sextuplets, septuplets, octuplets, nonuplets, and so on. I had to go to the full Oxford English Dictionary for the last one - the Concise Oxford jumps from 'octuplets' to 'decuplets'. The OED also defines 'dodecuplets', but apparently there are no words for any of the others. Presumably I could make them up, but instead I shall use the number itself when I want to refer to, for example, prime 11-tuplets. I couldn't find the general term 'k-tuplets' in the OED either, but it is the word that seems to be in common use by the mathematical community.
For now, I will define a prime k-tuplet as a sequence of consecutive prime numbers such that the distance between the first and the last is in some sense as small as possible. If you think I am being too vague, there is a more precise definition later on.
At this site I have collected together what I believe to be the largest known prime k-tuplets for k = 2, 3, 4, ..., 20 and 21. I do not know of any prime k-tuplets for k greater than 21, except for the ones that occur near the beginning of the prime number sequence.
Multiplication is often denoted by an asterisk: x*y is x times y.
For k > 2, the somewhat bizarre notation N + b1, b2, ..., bk is used (only in linked pages) to denote the k numbers {N + b1, N + b2, ..., N + bk}.
Prime twins are represented as N ± 1, which is short for N plus one and N minus one.
I also use the notation n# of Caldwell and Dubner [CD93] as a convenient shorthand for the product of all the primes less than or equal to n. Thus, for example, 20# = 2*3*5*7*11*13*17*19 = 9699690.
I would like to keep this site as up to date as possible. Therefore, can I urge you to please send any new, large prime k-tuplets to me. You can see what I mean by 'large' by studying the lists. If the numbers are not too big, say up to 500 digits, I am willing to double-check them myself. Otherwise I would appreciate some indication of how you proved that your numbers are true primes. Email address: anthony.d.forbes@gmail.com.
2996863034895 * 21290000 ± 1 (388342 digits, Sep 2016, Tom Greer, TWINGEN, PRIMEGRID, LLR)
3756801695685 * 2666669 ± 1 (200700 digits, Dec 2011, Timothy Winslow, TWINGEN, PRIMEGRID, LLR)
65516468355 * 2333333 ± 1 (100355 digits, Aug 2009, Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR)
70965694293 * 2200003 ± 1 (60219 digits, Apr 2016, S. Urushihata)
66444866235 * 2200003 ± 1 (60218 digits, Apr 2016, S. Urushihata)
4884940623 * 2198800 ± 1 (59855 digits, Jul 2015, Kwok, PSIEVE, LLR)
2003663613 * 2195000 ± 1 (58711 digits, Jan 2007, Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Michaek Kwok, Andrea Pacini, Rytis Slatkevicius)
38529154785 * 2173250 ± 1 (52165 digits, Jul 2014, Serge Batalov, NEWPGEN, LLR)
194772106074315 * 2171960 ± 1 (51780 digits, Jun 2007, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)
100314512544015 * 2171960 ± 1 (51780 digits, Dec 2006, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)
See Chris Caldwell, The Largest Known Primes for further (and possibly more up to date) information.
6521953289619 * 255555 + d, d = −5, −1, 1 (16737 digits, Apr 2013, Peter Kaiser)
3221449497221499 * 234567 + d, d = −1, 1, 5 (10422 digits, Sep 2015, Peter Kaiser, NEWGEN, LLR, PRIMO5)
1288726869465789 * 234567 + d, d = −5, −1, +1 (10421 digits, Apr 2014, Peter Kaiser)
81505264551807 * 233444 + d, d = −1, 1, 5 (10082 digits, Jul 2012, Peter Kaiser)
2072644824759 * 233333 + d, d = −1, 1, 5 (10047 digits, Nov 2008, Norman Luhn, François Morain, FastECPP [F09])
5612052289 * 14500# / 5 + d, d = −1, 1, 5 (6223 digits, Jan 2008, Norman Luhn, PRIMO)
(99241437759 * 205881 * 4001# (205881*4001# + 1) + 210) (205881*4001# − 1)/35 + d, d = 1, 5, 7 (5132 digits, Mar 2006, Ken Davis)
(91456744909 * 205881 * 4001# (205881*4001# + 1) + 210) (205881*4001# − 1)/35 + d, d = 5, 7, 11 (5132 digits, May 2006, Ken Davis)
(63140956174 * 205881 * 4001# (205881*4001# + 1) + 210) (205881*4001# − 1)/35 + d, d = 1, 5, 7 (5132 digits, Oct 2005, Ken Davis)
(63095588824 * 205881 * 4001# (205881*4001# + 1) + 210) (205881*4001# − 1)/35 + d, d = 7, 11, 13 (5132 digits, Oct 2005, Ken Davis)
4122429552750669 * 216567 + d, d = −1, 1, 5, 7 (5003 digits, Mar 2016, Peter Kaiser, GSIEVE, NewPGen, LLR, PRIMO)
2673092556681 * 153048 + d, d = −4, −2, 2, 4 (3598 digits, Sep 2015, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
2339662057597 * 103490 + d, d = 1, 3, 7, 9 (3503 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
305136484659 * 211399 + d, d = −1, 1, 5, 7 (3443 digits, Sep 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
722047383902589 * 211111 + d, d = −1, 1, 5, 7 (3360 digits, Apr 2013, Reto Keiser, NEWPGEN, PFGW, PRIMO)
43697976428649 * 29999 + d, d = −1, 1, 5, 7 (3024 digits, Mar 2012, Peter Kaiser)
46359065729523 * 28258 + d, d = −1, 1, 5, 7 (2500 digits, Nov 2011, Reto Keiser, NEWPGEN, PFGW, PRIMO)
1367848532291 * 5591# / 35 + d, d = −1, 1, 5, 7 (2401 digits, Aug 2011, Norman Luhn, NEWPGEN, PFGW, PRIMO)
25796119248 * 4987# / 35 + d, d = −1, 1, 5, 7 (2135 digits, May 2011, Gary Chaffey)
4104082046 * 4800# + 5651 + d, d = 0, 2, 6, 8 (2058 digits, Apr 2005, Norman Luhn, PRIMO)
2316765173284 * 3600# + 16061 + d, d = 0, 2, 6, 8, 12 (1543 digits, 16 Oct 2016, Norman Luhn, PRIMO)
163252711105 * 3371# / 2 + d, d = −8, −4, −2, 2, 4 (1443 digits, Jan 2014, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
9039840848561 * 3299# / 35 + d, d = −5, −1, 1, 5, 7 (1401 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
699549860111847 * 24244 + d, d = −1, 1, 5, 7, 11 (1293 digits, Dec 2013, Reto Keiser, R. Gerbicz, PFGW, PRIMO)
566650659276 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)
554729409262 * 2621# + 1615841 + d, d = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)
424232794973 * 2600# + 43777 + d, d = 0, 4, 6, 10, 12 (1107 digits, Mar 2009, Norman Luhn, PRIMO)
283534892623 * 2500# + 1091261 + d, d = 0, 2, 6, 8, 12 (1069 digits, Apr 2006, Norman Luhn)
96972480423104 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12 (1038 digits, Nov 2012, Norman Luhn, PRIMO)
91257770396975 * 2400# + 19421 + d, d = 0, 2, 6, 8, 12 (1037 digits, Aug 2012, Norman Luhn)
28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 (1037 digits, 14 Mar 2016, Norman Luhn, APSIEVE, PRIMO)
56955179868148127192911167380358423268347394410089842129140156658466369 * 1607# + 82984537 + d, d = 0, 4, 6, 10, 12, 16 (751 digits, Aug 2015, Vidar Nakling, PRIMO)
39192012090354765012032937869 63620446183193864072395084531670244391789850454987 90600310298931245078011665141488718221144402728008 84198554607462042723509866995108041645298700784996 91992151722541551955298630979698530841904537286500 37452220380184681391570092753684727879715473377537 * 1013# + 6503587 + d, d = 0, 4, 6, 10, 12, 16 (700 digits, Jan 2015, Vidar Nakling, PRIMO)
6897 02036532655186685581028503873005405874329363269153 97962209601434678501908870722030125604856836649860 28119644676547746708200919724631942081864768826993 86082393716593309811371422836387527549653095824492 75039409204553227509813565295242307835647237965390 89887138727590205662187634974598781067751832038576 48413997381256598543877696056491021898353604500233 20379862940392357016563411956474253654958412147188 16895693799643641522894946931181993379268860018434 60903637314310532482306798517536171711379098711480 66357226953506340768837768762395119697758299844912 09403588302768973281194836200119847131258596316036 52231485340570118364685553782567043880668996080767 + d, d = 0, 4, 6, 10, 12, 16 (654 digits, Nov 2014, the Riecoin project #164880, PRIMO)
4671972729725352818122941149688795 83036899825336211555483727688180233362871717424520 43367864623979884028668499412434355215700282247944 50613212179737967899963431468070908611907387424542 32569941618189864683903326675169723070698409551818 93752210505419805369423016027390095152729084370158 18063268988005118929691831130381608926396284126811 31149940006263513127749285237956463871166922433613 25227122575263902703163959067723823500146352052846 95255089488549517386620110262942368947858319697456 42070469581906895381174547631789334276066101601602 01941986359571297420634146813414364677014828568287 91215899277247997778189111505134511351782855366537 + d, d = 0, 4, 6, 10, 12, 16 (634 digits, Dec 2014, the Riecoin project #168912, PRIMO)
145706880639753656737279521958539 60515082637157543755159837625741219856420009823668 30244786215332121866228377649465535528368304607832 54125965129888309891835134016028148376391477335294 02861554469635099345877287765762209127962856755966 65491294239840306621145892093057985048351291074105 25330006954152630492665813319203174926300635633188 52682564644426662782037445701630356234114990357923 98375843863783926539038702211011242096871775954987 79757674767712364306682147365388176296110468770745 15443858755325749447345583441479831354111344777044 08334222893652785645775098226281002667790655225549 65904971178293533040332095883095722435581310387267 + d, d = 0, 4, 6, 10, 12, 16 (633 digits, 18 Nov 2014, the Riecoin project #160848, PRIMO)
870032513661292713201722972 60820165414512118488760301556398626498188491456463 58277686273968781999872868722459777111857484469717 53983280707629711154231572631053566230084276478860 23227207644547260993660031471135999127435197470213 13524857213838992173535645487774803822922600418333 62039710076475709098424235499491861853827976762822 55224158287242481391522282240117490588685419026728 * 547# + 7187767 + d, d = 0, 4, 6, 10, 12, 16 (600 digits, Oct 2014, Vidar Nakling, PRIMO)
492103509033322124528105 66004692030919102801408208961819403146196577563745 47612468071120273355418960561478350682547569140488 87521441845768296420138203277151345176983584805931 40810554258883403885165265601751315494217008739561 68222901685046046380856370538924059371410139041704 19238485529931649675829983229294105715547460874970 07967825846220015325654831770578435398632279854720 * 547# + 8061997 + d, d = 0, 4, 6, 10, 12, 16 (597 digits, Sep 2014, Vidar Nakling, PRIMO)
219946485329 * 1399# / 2 + d, d = −8, −4, −2, 2, 4, 8 (593 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
601545895935 * 1300# + 43777 + d, d = 0, 4, 6, 10, 12, 16 (559 digits, May 2009, Norman Luhn)
4733578067069 * 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20 (402 digits, May 2016, Norman Luhn)
4079068497377 * 739# / 14 + d, d = −4, −2, 2, 4, 8, 14, 16 (319 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)
771620215080738 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (305 digits, 4 Jul 2017, Norman Luhn, VFYPR)
375681809246516 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (305 digits, 4 Jul 2017, Norman Luhn, VFYPR)
359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
334063689033226 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
295031928451848 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
263868283973329 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
164124046378678 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
126443148358786 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)
29995576270632 * 550# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (236 digits, Jun 2014, Norman Luhn)
330846961 * 503# + 349129635971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (218 digits, Feb 2008, Jens Kruse Andersen)
12874261020824 * 465# + 88793 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 (206 digits, Aug 2005, Norman Luhn)
981580217350274 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)
860919759693785 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)
768132737748133 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)
663579549486449 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)
550907700932667 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)
527840551204869 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)
663579549486449 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (203 digits, 16 Mar 2017, Norman Luhn)
68663510211259 * 337# + 88789 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (150 digits, Jan 2010, Norman Luhn)
3336884 * 331# + 80877403191701 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (140 digits, Sep 2007, Dirk Augustin & Jens Kruse Andersen)
851437873414817 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (136 digits, 9 Feb 2017, Norman Luhn)
772556746441918 * 300# + 29247919 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 (136 digits, 9 Feb 2017, Norman Luhn)
772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (136 digits, 9 Feb 2017, Norman Luhn)
394833958615791 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 (135 digits, 9 Feb 2017, Norman Luhn)
106345403186416 * 300# + 29247913 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30 (135 digits, 9 Feb 2017, Norman Luhn)
90421624808713 * 300# + 103498931 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (135 digits, Feb 2005, Norman Luhn)
7644 * 281# + 355388175685385651 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (120 digits, May 2016, Roger Thompson)
772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (136 digits, 9 Feb 2017, Norman Luhn)
7425 * 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (120 digits, May 2016, Roger Thompson)
118557188915212 * 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (118 digits, Jun 2014, Norman Luhn)
24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, Aug 2004, Jens Kruse Andersen)
72613488698235 * 227# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (103 digits, Apr 2004, Norman Luhn)
613176722801194*151# + 177321217 + d, d = 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)
446098440707057*151# + 177321217 + d, d = 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)
422969185886875 * 151# + 177321223 + d, d = 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (74 digits, Aug 2014, Michael Stocker)
1587814371067375 * 150# + 23378471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (73 digits, Oct 2014, Norman Luhn)
1587447530513373 * 150# + 23378471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (73 digits, Oct 2014, Norman Luhn)
24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (104 digits, Aug 2004, Norman Luhn & Jens Kruse Andersen)
613176722801194*151# + 177321217 + d, d = 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)
446098440707057*151# + 177321217 + d, d = 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)
422969185886875 * 151# + 177321223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (74 digits, Aug 2014, Michael Stocker)
705484555578416 * 150# + 23378471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (73 digits, Oct 2014, Norman Luhn)
35078052 * 157# + 398861548425071 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)
34101658 * 157# + 164826429367331 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)
467756 * 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (66 digits, May 2014, Roger Thompson)
10015646838 * 139# + 2610065603261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (66 digits, Aug 2013, Roger Thompson)
92119245478633 * 130# + 21816911 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (63 digits, Dec 2003, Norman Luhn)
613176722801194*151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)
467756 * 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (66 digits, May 2014, Roger Thompson)
59125383480754 * 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (61 digits, Sep 2013, Michael Stocker)
78989413043158 * 109# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (59 digits, Jan 2010, Norman Luhn)
450725899 * 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (56 digits, Nov 2014, Martin Raab)
14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, Feb 2013, Roger Thompson)
8486221 * 107# + 4549290807806861 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)
839858 * 107# + 2566964683459061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (49 digits, Aug 2009, Jens Kruse Andersen)
337712 * 107# + 3440354553191441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (48 digits, Aug 2009, Jens Kruse Andersen)
122428 * 107# + 4540852852571921 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (48 digits, Aug 2009, Jens Kruse Andersen)
4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (61 digits, 23 Mar 2017, Norman Luhn)
14815550 * 107# + 4385574275277311 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (50 digits, Feb 2013, Roger Thompson)
14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (50 digits, Feb 2013, Roger Thompson)
61571 * 107# + 4803194122972361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (48 digits, Aug 2009, Jens Kruse Andersen)
381955327397348*80# + 18393209 + d, d = 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)
381955327397348*80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (46 digits, Dec 2007, Norman Luhn)
1955206838 * 73# + 44208109063 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (38 digits, Aug 2012, Martin Raab)
322255 * 73# + 1354238543317302647 + d, d = 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (35 digits, 18 Nov 2016, Roger Thompson)
1464893944 * 67# + 42166182984041 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (35 digits, Jul 2012, Martin Raab)
457308940 * 67# + 4122369405991 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 (34 digits, Mar 2011, Martin Raab)
14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (50 digits, Feb 2013, Roger Thompson)
381955327397348*80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)
26093748 * 67# + 383123187762431 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
108804167016152508211944400342691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
101885197790002105359911556070661 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
101803109763079694387921584406441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
101047123513223569167212934432341 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
100859765410802682029505696121301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
100496797396678760339871075201851 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)
33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (40 digits, 25 Jan 2017, Norman Luhn)
322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (35 digits, 18 Nov 2016, Roger Thompson)
10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (35 digits, Sep 2012, Roger Thompson)
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (33 digits, Apr 2008, Jens Kruse Andersen)
99999999948164978600250563546400 + d, d = 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67 (32 digits, Nov 2004, Joerg Waldvogel and Peter Leikauf)
1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)
1100916249233879857334075234831 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)
1003234871202624616703163933853 + d, d = 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)
999999999962618227626700812281 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (30 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)
100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (30 digits, Feb 2013, Roger Thompson)
322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (35 digits, 18 Nov 2016, Roger Thompson)
1003234871202624616703163933853 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (29 digits, Jan 2012, Roger Thompson)
5867208169546174917450987997 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4652363394518920290108071167 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4483200447126419500533043987 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
3361885098594416802447362317 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
3261917553005305074228431077 + d, d = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (30 digits, Feb 2013, Roger Thompson)
11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (29 digits, Jan 2012, Roger Thompson)
11410793439953412180643704677 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (29 digits, Jan 2012, Roger Thompson)
5867208169546174917450987997 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4652363394518920290108071167 + d, d = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4652363394518920290108071167 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
4483200447126419500533043987 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)
3361885098594416802447362317 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
3261917553005305074228431077 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
3176488693054534709318830357 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)
2650778861583720495199114537 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Feb 2013, Raanan Chermoni & Jaroslaw Wroblewski)
2406179998282157386567481191 + d, d = 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)
248283957683772055928836513589 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)
138433730977092118055599751669 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski)
39433867730216371575457664399 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (29 digits, 8 Jan 2015, Raanan Chermoni & Jaroslaw Wroblewski)
2406179998282157386567481191 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)
2348190884512663974906615481 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)
917810189564189435979968491 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, May 2011, Raanan Chermoni & Jaroslaw Wroblewski)
656632460108426841186109951 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, 19 Feb 2011, Raanan Chermoni & Jaroslaw Wroblewski)
630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, 9 Feb 2011, Raanan Chermoni & Jaroslaw Wroblewski)
{37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
{13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89}
352259532126245901475150592651 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, 24 May 2017, Raanan Chermoni & Jaroslaw Wroblewski)
349021296319127268299400177269 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, 24 May 2017, Raanan Chermoni & Jaroslaw Wroblewski)
320430661688896578454772807699 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, 24 May 2017, Raanan Chermoni & Jaroslaw Wroblewski)
260786413629117930695179308299 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, 7 Dec 2016, 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 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)
242145669497919182306126385461 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)
227104691557231224024329351201 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)
220974232729147341120519932981 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, 31 May 2016, Raanan Chermoni & Jaroslaw Wroblewski)
187976201367296936422347098471 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, 16 Mar 2016, Raanan Chermoni & Jaroslaw Wroblewski)
185986500598659638316208079201 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, 16 Mar 2016, Raanan Chermoni & Jaroslaw Wroblewski)
248283957683772055928836513589 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)
138433730977092118055599751669 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski)
39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (29 digits, 8 Jan 2015, Raanan Chermoni & Jaroslaw Wroblewski)
{29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
| The largest known prime k-tuplets | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 22338618 | 274207281 − 1 | Curtis Cooper, George Woltman, Scott Kurowski et al (GIMPS) | 7 Jan 2016 |
| 2 | 388342 | 2996863034895 * 21290000 ± 1 | Tom Greer, TWINGEN, PRIMEGRID, LLR | Sep 2016 |
| 3 | 16737 | 6521953289619 * 255555 + d, d = −5, −1, 1 | Peter Kaiser | Apr 2013 |
| 4 | 5003 | 4122429552750669 * 216567 + d, d = −1, 1, 5, 7 | Peter Kaiser, GSIEVE, NewPGen, LLR, PRIMO | Mar 2016 |
| 5 | 1543 | 2316765173284 * 3600# + 16061 + d, d = 0, 2, 6, 8, 12 | Norman Luhn, PRIMO | 16 Oct 2016 |
| 6 | 1037 | 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 | Norman Luhn, APSIEVE, PRIMO | 14 Mar 2016 |
| 7 | 402 | 4733578067069 * 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20 | Norman Luhn | May 2016 |
| 8 | 304 | 359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | Norman Luhn, VFYPR | 4 Jul 2017 |
| 9 | 203 | 663579549486449 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 | Norman Luhn | Mar 2017 |
| 10 | 136 | 772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 | Norman Luhn | 9 Feb 2017 |
| 11 | 104 | 24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 | Norman Luhn & Jens Kruse Andersen | Aug 2004 |
| 12 | 75 | 613176722801194*151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 | Michael Stocker, PRIMO | Sep 2014 |
| 13 | 61 | 4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 | Norman Luhn | 23 Mar 2017 |
| 14 | 50 | 14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | Roger Thompson | Feb 2013 |
| 15 | 40 | 33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 | Norman Luhn | 25 Jan 2017 |
| 16 | 35 | 322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | Roger Thompson | 18 Nov 2016 |
| 17 | 30 | 100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 | Roger Thompson | Feb 2013 |
| 18 | 28 | 5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | Raanan Chermoni & Jaroslaw Wroblewski | Mar 2014 |
| 19 | 30 | 248283957683772055928836513589 + d, d = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 | Raanan Chermoni & Jaroslaw Wroblewski | 1 Aug 2016 |
| 20 | 30 | 352259532126245901475150592651 + 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 | 24 May 2017 |
| 21 | 30 | 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 | 1 Aug 2016 |
| 22 | 2 | {23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |
| 23 | 2 | {19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |
| 24 | - | There are no known prime 24-tuplets | - | - |
| First appearance of a non-trivial prime k-tuplet | ||||
| k | Digits | Prime k-tuplet | Who | When |
| <12 | - | No reliable information | - | - |
| 12 | 13 | 1418575498567 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 | D. Betsis & S. Säfholm | 1982 |
| 13 | 14 | 10527733922579 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 | D. Betsis & S. Säfholm | 1982 |
| 14 | 17 | 21817283854511261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | D. Betsis & S. Säfholm | 1982 |
| 15 | 21 | 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 | Joerg Waldvogel | 1996 |
| 16 | 21 | 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | Joerg Waldvogel | 1996 |
| 17 | 22 | 1620784518619319025971 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | Joerg Waldvogel | 1997 |
| 18 | 25 | 2845372542509911868266807 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | Joerg Waldvogel & Peter Leikauf | 14 Nov 2000 |
| 19 | 27 | 630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 | Raanan Chermoni & Jaroslaw Wroblewski | 9 Feb 2011 |
| 20 | 28 | 3941119827895253385301920029 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | Raanan Chermoni & Jaroslaw Wroblewski | 24 Jun 2014 |
| 21 | 29 | 39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 | Raanan Chermoni & Jaroslaw Wroblewski | 8 Jan 2015 |
| First appearance of 100 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 157 | 2521 − 1 | R. M. Robinson | Jan 1952 |
| 2-5 | - | No reliable information | - | - |
| 6 | 133 | 2 * 10132 + 75543532187 + d, d = 0, 4, 6, 10, 12, 16 | TF | Apr 1994 |
| 7 | 104 | 4293326603 * 233# + 399389 + d, d = 0, 2, 8, 12, 14, 18, 20 | Radoslaw Naleczynski | Dec 1998 |
| 8 | 110 | 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | Norman Luhn | Feb 2001 |
| 9 | 110 | 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 | Norman Luhn | Feb 2001 |
| 10 | 103 | 72613488698235 * 227# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 | Norman Luhn | Apr 2004 |
| 11 | 104 | 24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 | Norman Luhn & Jens Kruse Andersen | Aug 2004 |
| First appearance of 1000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 1332 | 24423 − 1 | Alexander Hurwitz | Nov 1961 |
| 2 | 1040 | 256200945 * 23426 ± 1 | Oliver Atkin & N. W. Rickert | 1980 |
| 3 | 1083 | 437850590*(23567 − 21189) − 6*21189 + d, d = −5, −1, 1 | Tony Forbes | Dec 1996 |
| 4 | 1004 | 76912895956636885*(23279 − 21093) − 6*21093 + d, d = −7, −5, −1, 1 | Tony Forbes | Sep 1998 |
| 5 | 1034 | 31969211688*2400# + 16061 + d, d = 0, 2, 6, 8, 12 | Norman Luhn | Jul 2002 |
| 6 | 1037 | 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 | Norman Luhn, APSIEVE, PRIMO | 14 Mar 2016 |
| First appearance of 10000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 13395 | 244497 − 1 | Harry Nelson & David Slowinski | Apr 1979 |
| 2 | 11713 | 242206083 * 238880 ± 1 | H. K. Indlekofer & A. Járai | Nov 1995 |
| 3 | 10047 | 2072644824759 * 233333 + d, d = −1, 1, 5 | Norman Luhn, François Morain, FastECPP | Nov 2008 |
| First appearance of 100000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 227832 | 2756839 − 1 | David Slowinski & Paul Gage | Apr 1992 |
| 2 | 100355 | 65516468355 * 2333333 ± 1 | Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR | Aug 2009 |
| First appearance of 1000000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 2098960 | 26972593 − 1 | Nayan Hajratwala, George Woltman, Scott Kurowski et al (GIMPS) | Jun 1999 |
| First appearance of 10000000 digits | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 12978189 | 243112609 − 1 | Edson Smith, George Woltman, Scott Kurowski et al (GIMPS) | Sep 2008 |
List of all possible patterns of prime k-tuplets
List of the smallest prime k-tuplets
Near misses: Clusters of primes that didn't quite make it into the main list
The Hardy-Littlewood constants pertaining to the distribution of prime k-tuplets [HL22]
Jens Kruse Andersen: The Largest Known Simultaneous Primes
Jens Kruse Andersen: Consecutive Primes in Arithmetic Progression
Jens Kruse Andersen: Largest Consecutive Factorizations
Dirk Augustin: Cunningham Chain records
Chris K. Caldwell: The Largest Known Primes
Chris K. Caldwell: Top twenty twin primes
TF: Ten consecutive primes in arithmetic progression
Dr. Minh. L. Perez Press: Smarandache Primes
N. J. A. Sloane: On-Line Encyclopedia of Integer Sequences
Manfred Toplic: The Nine and Ten Primes Project
Robin Whitty: Theorem of the Day
A prime k-tuplet is a sequence of k consecutive prime numbers such that in some sense the difference between the first and the last is as small as possible. The idea is to generalise the concept of prime twins.
More precisely: We first define s(k) to be the smallest number s for which there exist k integers b1 < b2 < ... < bk, bk − b1 = s and, for every prime q, not all the residues modulo q are represented by b1, b2, ..., bk. A prime k-tuplet is then defined as a sequence of consecutive primes {p1, p2, ..., pk} such that for every prime q, not all the residues modulo q are represented by p1, p2, ..., pk, pk − p1 = s(k). Observe that the definition excludes a finite number (for each k) of dense clusters at the beginning of the prime number sequence - for example, {97, 101, 103, 107, 109} satisfies the conditions of the definition of a prime 5-tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.
The simplest case is s(2) = 2, corresponding to prime twins: {p, p + 2}. Next, s(3) = 6 and two types of prime triplets: {p, p + 2, p + 6} and {p, p + 4, p + 6}, followed by s(4) = 8 with just one pattern: {p, p + 2, p + 6, p + 8} of prime quadruplets. The sequence continues with s(5) = 12, s(6) = 16, s(7) = 20, s(8) = 26, s(9) = 30, s(10) = 32, s(11) = 36, s(12) = 42, s(13) = 48, s(14) = 50, s(15) = 56, s(16) = 60, s(17) = 66 and so on. It is number A008407 in N.J.A. Sloane's On-line Encyclopedia of Integer Sequences.
In keeping with similar published lists, I have decided not to accept anything other than true, proven primes. Numbers which have merely passed the Fermat test, aN = a (mod N), will need to be validated. If N − 1 or N + 1 is sufficiently factorized (usually just under a third), the methods of Brillhart, Lehmer and Selfridge [BLS75] will suffice. Otherwise the numbers may have to be subjected to a general primality test, such as the Jacobi sum test of Adleman, Pomerance, Rumely, Cohen and Lenstra (APRT-CLE in UBASIC, for example), or one of the elliptic curve primality proving programs: Atkin and Morain's ECPP, or its successor, Franke, Kleinjung, Wirth and Morain's FAST-ECPP, or Marcel Martin's PRIMO.
Euclid proved that there are infinitely many primes. Paulo Ribenboim [Rib95] has collected together a considerable number of different proofs of this important theorem. My favourite (which is not in Ribenboim's book) goes like this: We have
∏p prime 1/(1 − 1/p2) = ∑n = 1 to ∞ 1/n2 = π2/6.
But π2 is irrational, so the product on the left cannot have a finite number of factors.
In its simplest form, the prime number theorem states that the number of primes less than x is asymptotic to x/(log x). This was first proved by Hadamard and independently by de la Vallee Poussin in 1896. Later, de la Vallee Poussin found a better estimate:
∫u = 0 to x du/(log u) + error term,
where the error term is bounded above by A x exp(−B √(log x)) for some constants A and B. With more work (H.-E. Richert, 1967), √(log x) in this last expression can be replaced by (log x)3/5(log log x)−1/5. The most important unsolved conjecture of prime number theory, indeed, all of mathematics - the Riemann Hypothesis - asserts that the error term can be bounded by a function of the form 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 C2 x / (log x)2,
where C2 = ∏p prime, p > 2 p(p − 2) / (p − 1)2 = 0.66016... is known as the twin prime constant.
V. Brun showed that the sequence of twins is thin enough for the series ∑p and p + 2 prime 1 / p to converge. The twin prime conjecture states that the sum has infinitely many terms. The nearest to proving the conjecture is Jing-Run Chen's result that there are infinitely many primes p such that p + 2 is either prime or the product of two primes [HR73].
The Partitio Numerorum: III paper [HL22] goes on to formulate a general conjecture concerning the distribution of arbitrary groups of prime numbers (The k-tuplets of this site are special cases): Let b1, b2, ..., bk be k distinct integers. Then the number of groups of primes N + b1, N + b2, ..., N + bk between 2 and x is approximately
Hk Ck ∫u = 2 to x du / (log u)k,
where
Hk = ∏p prime, p ≤ k pk − 1 (p − v) / (p − 1)k ∏p prime, p > k, p|D (p − v) / (p − k),
Ck = ∏p prime, p > k pk − 1 (p − k) / (p − 1)k,
v = v(p) is the number of distinct remainders of b1, b2, ..., bk modulo p and D is the product of the differences |bi − bj|, 1 ≤ i < j ≤ k.
The first product in Hk is over the primes not greater than k, the second is over the primes greater than k which divide D and the product Ck is over all primes greater than k. If you put k = 2, b1 = 0 and b2 = 2, then v(2) = 1, v(p) = p − 1 for p > 2, H2 = 2, and Ck = C2, the twin prime constant given above.
It is worth pointing out that with modern mathematical software the prime k-tuplet constants Ck can be determined to great accuracy. The way not to do it is to use the defining formula. Unless you are very patient, calculating the product over a sufficient number of primes for, say, 20 decimal place accuracy would not be feasible. Instead there is a useful transformation originating from the product formula for the Riemann ζ function:
log Ck = − ∑n = 2 to ∞ log [ζ(n) ∏p prime, p ≤ k (1 − 1/pn)] / n ∑d|n μ(n/d) (kd − k).
[BLS75] John Brillhart, D.H. Lehmer & J.L. Selfridge, New primality criteria and factorizations of 2m ± 1, Math. Comp. 29 (1975), 620-647.
[CD93] C.K.Caldwell & H. Dubner, Primorial, factorial and multifactorial primes, Math. Spectrum 26 (1993/94), 1-7.
[F97f] Tony Forbes, Prime 17-tuplet, NMBRTHRY Mailing List, September 1997.
[F02] Tony Forbes, Titanic prime quintuplets, M500 189 (December, 2002), 12-13.
[F09] Tony Forbes, Gigantic prime triplets, M500 226 (February, 2009), 18-19.
[Guy94] Richard K. Guy, Unsolved Problems in Number Theory, second edn., Springer-Verlag, New York 1994.
[HL22] G. H. Hardy and J. E. Littlewood, Some problems of Partitio Numerorum: III; on the expression of a number as a sum of primes, Acta Mathematica 44 (1922), 1-70.
[HR73] H. Halberstam and H.-E Richert, Sieve Methods, Academic Press, London 1973.
[Rib95] P. Ribenboim, The New Book of Prime Number Records, 3rd edn., Springer-Verlag, New York 1995
[R95] Warut Roonguthai, Prime quadruplets, NMBRTHRY Mailing List, September 1995.
[R96a] Warut Roonguthai, Prime quadruplets, M500 148 (February 1996), 9.
[R96b] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1996.
[R96c] Warut Roonguthai, Large prime quadruplets, M500, 153 (December, 1996), 4-5.
[R97a] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1997.
[R97b] Warut Roonguthai, Large prime quadruplets, M500 158 (November 1997), 15.
[S95] G. John Stevens, Prime Quadruplets, J. Recr. Math. 27 (1995), 17-22.