At this site we collect 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.
Last updated: 14 April 2008.
Site address: http://anthony.d.forbes.googlepages.com/ktuplets.htm.
Old address: http://www.ltkz.demon.co.uk/ktuplets.htm.
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. The three numbers will always include a multiple of 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, ..., 17 and 18. I do not know of any prime k-tuplets for k greater than 18, 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.
2003663613 * 2195000 ± 1 (58711 digits, Jan 2007, Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Michaek Kwok, Andrea Pacini, Rytis Slatkevicius)
194772106074315 * 2171960 ± 1 (51780 digits, June 2007, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)
100314512544015 * 2171960 ±1 (51780 digits, Dec 2006, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)
16869987339975 * 2171960 ± 1 (51779 digits, Sep 2005, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)
33218925 * 2169690 ± 1 (51090 digits, 2002, Daniel Papp & Yves Gallot)
60194061 * 2114689 ± 1 (34533 digits, 2002, David Underbakke)
1765199373 * 2107520 ± 1 (32376 digits, 2002, James McElhatton & Yves Gallot)
318032361 * 2107001 ± 1 (32220 digits, 2001, David Underbakke & Phil Carmody)
1046619117*2100000 ± 1 (30113 digits, Oct 2007, Gary Barnes)
1807318575 * 298305 ± 1 (29603 digits, 2001, David Underbakke & Phil Carmody)
See Chris Caldwell, The Largest Known Primes, for further (and possibly more up-to-date) information.
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)
(62258488321368 * 3331# * (1037*3331#+1) + 210) (1037*3331# − 1)/35 + d, d = 5, 7, 11 (4259 digits, Jul 2003, David Broadhurst)
(22877907949788 * 3331# * (1037*3331#+1) + 210) (1037*3331# − 1)/35 + d, d = 5, 7, 11 (4259 digits, Jul 2003, David Broadhurst)
(108748629354 * 4436 * 3251# * (4436*3251# + 1) + 210) (4436*3251# − 1)/35 + d, d = 7, 11, 13 (4135 digits, Sep 2002, David Broadhurst, PFGW)
(90159302514 * 4436 * 3251# * (4436*3251# + 1) + 210) (4436*3251# − 1)/35 + d, d = 5, 7, 11 (4135 digits, Sep 2002, David Broadhurst, PFGW)
(39553075974 * 4436 * 3251# * (4436*3251# + 1) + 210) (4436*3251# − 1)/35 + d, d = 5, 7, 11 (4135 digits, Sep 2002, David Broadhurst, PFGW)
4104082046 * 4800# + 5651 + d, d = 0, 2, 6, 8 (2058 digits, Apr 2005, Norman Luhn, PRIMO)
11024895887 * 3500# + 855731 + d, d = 0, 2, 6, 8 (1491 digits, Feb 2003, Norman Luhn, PRIMO)
10271674954 * 2999# + 3461 + d, d = 0, 2, 6, 8 (1284 digits, Feb 2002, Michael Bell, Michael Davison, Matt Jack, Ronald Lau, Graeme Leese and Ben Lowing)
2722420456827 * 23800 + d, d = −1, 1, 5, 7 (1157 digits, Oct 2007, Gary Barnes)
477707955423 * 23802 + d, d = −1, 1, 5, 7 (1157 digits, Oct 2007, Gary Barnes)
111101 + 45917626999140 + d, d = 0, 2, 6, 8 (1147 digits, Sep 2005, Ronny Edler)
111101 + 34264768249680 + d, d = 0, 2, 6, 8 (1147 digits, Sep 2005, Ronny Edler)
700209251206546 * 32239 + d, d = −1, 1, 5, 7 (1084 digits, Aug 2005, Michael Gillion & George Woltman)
18973472837 * 2503#/35 + d, d = −1, 1, 5, 7 (1070 digits, Aug 2005, Gary Chaffey, PRIMO)
283534892623 * 2500# + 1091261 + d, d = 0, 2, 6, 8 (1068 digits, Apr 2006, Norman Luhn)
283534892623 * 2500# + 1091261 + d, d = 0, 2, 6, 8, 12 (1069 digits, Apr 2006, Norman Luhn)
31969211688 * 2400# + 16061 + d, d = 0, 2, 6, 8, 12 (1034 digits, Jul 2002, Norman Luhn [F02], APSIEVE, PFGW, PRIMO)
912143859 * 1223# + 463001711 + d, d = 0, 2, 6, 8, 12 (522 digits, Mar 2004, Donovan Johnson)
19685846183 * 1200# + 6005891 + d, d = 0, 2, 6, 8, 12 (511, digits, May 2002, Norman Luhn, PFGW, PRIMO)
14519751105 * 1050# + 1042090781 + d, d = 0, 2, 6, 8, 12 (450 digits, Apr 2002, Michael Hannigan)
338769039776 * 1000# + 16061 + d, d = 0, 2, 6, 8, 12 (427 digits, Jan 2006, Norman Luhn)
328481121285 * 1000# + 16057 + d, d = 4, 6, 10, 12, 16 (427 digits, Jan 2006, Norman Luhn)
328481121285 * 1000# + 16057 + d, d = 0, 4, 6, 10, 12 (427 digits, Jan 2006, Norman Luhn)
319335512503 * 1000# + 16061 + d, d = 0, 2, 6, 8, 12 (427 digits, Jan 2006, Norman Luhn)
305727029371 * 1000# + 16061 + d, d = 0, 2, 6, 8, 12 (427 digits, Jan 2006, Norman Luhn)
328481121285 * 1000# + 16057 + d, d = 0, 4, 6, 10, 12, 16 (427 digits, Jan 2006, Norman Luhn)
138765468778 * 850# + 2822707 + d, d = 0, 4, 6, 10, 12, 16 (362 digits, Apr 2004, Norman Luhn)
8398544501 * 710# + 2000472907 + d, d = 0, 4, 6, 10, 12, 16 (306 digits, Aug 2003, Torbjörn Alm & Jens Kruse Andersen, VFYPR)
110282080125 * 700# + 6005887 + d, d = 0, 4, 6, 10, 12, 16 (301 digits, Oct 2001, Norman Luhn, PRIMO)
97953153175 * 670# + 16057 + d, d = 0, 4, 6, 10, 12, 16 (290 digits, Apr 2001, Michael Bell, Graeme Leese, Michael Davison, APSIEVE, TITANIX)
86450022463 * 570# + 1000000587445747 + d, d = 0, 4, 6, 10, 12, 16 (242 digits, Jan 2001, Norman Luhn)
1189609319 * 503#/613777 + 446215867 + d, d = 0, 4, 6, 10, 12, 16 (213 digits, Sep 2000, Michael Bell)
82248305245 * 43# * 2479 + 16057 + d, d = 0, 4, 6, 10, 12, 16 (172 digits, 1997, A.O.L. Atkin)
10160 + 72849172960797 + d, d = 0, 4, 6, 10, 12, 16 (161 digits, Oct 2005, Richard Miller)
2512 + 6638977280721 + d, d = 0, 4, 6, 10, 12, 16 (155 digits, 1996, Tony Forbes [F96f])
251733155478 * 650# + 1146779 + d, d = 0, 2, 8, 12, 14, 18, 20 (282 digits, Jan 2006, Norman Luhn)
76794640264 * 509# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20 (223 digits, Sep 2004, Jens Kruse Andersen)
1839198074074 * 500# + 165701 + d, d = 0, 2, 6, 8, 12, 18, 20 (219 digits, Jun 2004, Norman Luhn)
581889703 * 503# + 138264351341 + d, d = 0, 2, 6, 8, 12, 18, 20 (218 digits, Feb 2008, Jens Kruse Andersen)
456986419 * 503# + 161951504831 + d, d = 0, 2, 6, 8, 12, 18, 20 (218 digits, Feb 2008, Jens Kruse Andersen)
428040647 * 503# + 29941937141 + d, d = 0, 2, 6, 8, 12, 18, 20 (218 digits, Feb 2008, Jens Kruse Andersen)
330846961 * 503# + 349129635971 + d, d = 0, 2, 6, 8, 12, 18, 20 (218 digits, Feb 2008, Jens Kruse Andersen)
315131450 * 503# + 108685942481 + d, d = 0, 2, 6, 8, 12, 18, 20 (218 digits, Feb 2008, Jens Kruse Andersen)
230249877 * 503# + 302416919921 + d, d = 0, 2, 6, 8, 12, 18, 20 (218 digits, Feb 2008, Jens Kruse Andersen)
221801776 * 503# + 33874245521 + d, d = 0, 2, 6, 8, 12, 18, 20 (218 digits, Feb 2008, Jens Kruse Andersen)
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)
97510235 * 421# + 322355908991 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (180 digits, Jan 2005, Torbjörn Alm & Jens Kruse Andersen)
4319152256906 * 400# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (174 digits, Nov 2004, Norman Luhn)
65677369861 * 380# + 18000020393471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (165 digits, Aug 2003, Torbjörn Alm & Jens Kruse Andersen)
15234072433401 * 375# + 43813839521 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (164 digits, May 2002, Norman Luhn)
243551752728 * 320# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26 (142 digits, Jun 2001, Graeme Leese, Michael Bell, Matt Jack, Michael Davison, Ben Lowing, Tim Nightingale, APSieve)
15040372 * 331# + 24632288535971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (140 digits, Sep 2007, Dirk Augustin & Jens Kruse Andersen)
14167183 * 331# + 35015517882461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (140 digits, Sep 2007, Dirk Augustin & Jens Kruse Andersen)
13603476 * 331# + 55083998744891 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 (140 digits, Sep 2007, Dirk Augustin & Jens Kruse Andersen)
3336884 * 331# + 80877403191701 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (140 digits, Sep 2007, Dirk Augustin & Jens Kruse Andersen)
90421624808713 * 300# + 103498931 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (135 digits, Feb 2005, Norman Luhn)
1619062142 * 255# + 53344165991 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (111 digits, Aug 2003, Thomas Wolter & Jens Kruse Andersen)
388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (110 digits, Feb 2001, Norman Luhn)
2242445342405 * 230# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Jun 2003, Norman Luhn)
24698258 * 239# + 28606476153371 + d, d = 6, 8, 12, 18, 20, 26, 30, 32, 36 (104 digits, Aug 2004, Norman Luhn & Jens Kruse Andersen)
24698258 * 239# + 28606476153371 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, Aug 2004, Jens Kruse Andersen)
24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Aug 2004, Jens Kruse Andersen)
18188893 * 239# + 7597110015611 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Aug 2004, Jens Kruse Andersen)
12023698 * 239# + 19623646397471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Aug 2004, Jens Kruse Andersen)
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)
36273553 * 157# + 106263743005151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)
35405054 * 157# + 143751012544871 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)
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 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)
33744069 * 157# + 243858308984021 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)
33395983 * 157# + 49822093470881 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)
31812413 * 157# + 394317153630131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)
30972388 * 157# + 218344297501061 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)
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)
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)
92119245478633 * 130# + 21816911 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (63 digits, Dec 2003, Norman Luhn)
58187756 * 110# + 2320048690691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (53 digits, Aug 2003, Jens Kruse Andersen)
11450665899501 * 101# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (52 digits, Aug 2003, Norman Luhn)
8747677 * 107# + 4008289033283651 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)
8519951 * 107# + 4108713619105211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)
8486221 * 107# + 4549290807806861 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)
7862516 * 107# + 4453056384461801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)
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)
381955327397348*80# + 18393209 + d, d = 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)
434161298 * 89# + 612442658382671 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (44 digits, Dec 2004, Christ van Willegen & Jens Kruse Andersen)
432589236 * 89# + 611531575179641 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (44 digits, Dec 2004, Christ van Willegen & Jens Kruse Andersen)
332352838383 * 80# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (43 digits, Jul 2003, Jens Kruse Andersen)
5027317106963 * 75# + 1418575498567 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (42 digits, Nov 2001, Norman Luhn)
720345861287087 * 70# + 8393501 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)
564115572162757 * 70# + 8393501 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)
412477355651067 * 70# + 8393501 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)
186519833921143 * 70# + 8393501 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)
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)
26697593 * 67# + 315911634133211 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
26307518 * 67# + 184083066052001 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
26093748 * 67# + 383123187762431 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
26093748 * 67# + 383123187762431 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
25478069 * 67# + 114181562199821 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
24970179 * 67# + 164226895277561 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
23320603 * 67# + 60301221485621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
22996461 * 67# + 257514231089231 + d, d = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)
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)
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)
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)
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)
10252256693298561414756287 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (26 digits, Oct 2004, Jens Kruse Andersen)
2845372542509911868266817 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (25 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)
1906230835046648293290047 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (25 digits, Feb 2001, Joerg Waldvogel & Peter Leikauf)
163027495131423420474917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (24 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
123452114023762529883167 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (24 digits, Jul 1999, Joerg Waldvogel)
10252256693298561414756287 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (26 digits, Oct 2004, Jens Kruse Andersen)
2845372542509911868266817 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (25 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)
1906230835046648293290043 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (25 digits, Feb 2001, Joerg Waldvogel & Peter Leikauf)
163027495131423420474913 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (24 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
110885131130067570042703 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (24 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
83405687980406998933663 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (23 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
78314167738064529047713 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (23 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
47624415490498763963983 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (23 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
20947353617877810296177 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (23 digits, Mar 1999, Tony Forbes)
3259125690557440336637 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (22 digits, Sep 1997, Tony Forbes [F97f])
2845372542509911868266811 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 (25 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)
2845372542509911868266807 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (25 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)
1906230835046648293290047 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (25 digits, Feb 2001, Joerg Waldvogel & Peter Leikauf)
1906230835046648293290043 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (25 digits, Feb 2001, Joerg Waldvogel & Peter Leikauf)
163027495131423420474913 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (24 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
110885131130067570042703 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (24 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
83405687980406998933663 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (23 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
78314167738064529047713 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (23 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
53947453971035573715707 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (23 digits, Aug 1998, Tony Forbes)
47624415490498763963983 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (23 digits, May 2001, Peter Leikauf & Joerg Waldvogel)
2845372542509911868266807 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (25 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)
1906230835046648293290043 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66, 70 (25 digits, Feb 2001, Joerg Waldvogel & Peter Leikauf)
{13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83}
| The largest known prime k-tuplets | ||||
| k | Digits | Prime k-tuplet | Who | When |
| 1 | 9808358 | 232582657 − 1 | Curtis Cooper, Steven Boone, George Woltman, Scott Kurowski, et al. | Sep 2006 |
| 2 | 58711 | 2003663613 * 2195000 ± 1 | Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Michaek Kwok, Andrea Pacini, Rytis Slatkevicius | Jan 2007 |
| 3 | 6223 | 5612052289 * 14500# / 5 + d, d = −1, 1, 5 | Norman Luhn, PRIMO | Jan 2008 |
| 4 | 2058 | 4104082046 * 4800# + 5651 + d, d = 0, 2, 6, 8 | Norman Luhn, PRIMO | Apr 2005 |
| 5 | 1069 | 283534892623 * 2500# + 1091261 + d, d = 0, 2, 6, 8, 12 | Norman Luhn | Apr 2006 |
| 6 | 427 | 328481121285 * 1000# + 16057 + d, d = 0, 4, 6, 10, 12, 16 | Norman Luhn | Jan 2006 |
| 7 | 282 | 251733155478 * 650# + 1146779 + d, d = 0, 2, 8, 12, 14, 18, 20 | Norman Luhn | Jan 2006 |
| 8 | 218 | 330846961 * 503# + 349129635971 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | Jens Kruse Andersen | Feb 2008 |
| 9 | 140 | 3336884 * 331# + 80877403191701 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 | Dirk Augustin & Jens Kruse Andersen | Sep 2007 |
| 10 | 104 | 24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 | Jens Kruse Andersen | Aug 2004 |
| 11 | 104 | 24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 | Norman Luhn & Jens Kruse Andersen | Aug 2004 |
| 12 | 50 | 8486221 * 107# + 4549290807806861 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | Dirk Augustin & Jens Kruse Andersen | May 2006 |
| 13 | 46 | 381955327397348*80# + 18393209 + d, d = 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 | Norman Luhn | Dec 2007 |
| 14 | 46 | 381955327397348*80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 | Norman Luhn | Dec 2007 |
| 15 | 33 | 107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 | Jens Kruse Andersen | Apr 2008 |
| 16 | 26 | 10252256693298561414756287 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | Jens Kruse Andersen | Oct 2004 |
| 17 | 25 | 2845372542509911868266811 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | Joerg Waldvogel & Peter Leikauf | Nov 2000 |
| 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 | Nov 2000 |
| 19 | 2 | {37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |
| 20 | 2 | {29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109} | - | - |
| 21 | 2 | {29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |
| 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 | - | - |
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
Y. H. Huen: Goldbach Sequences
Dr. Minh. L. Perez Press: Smarandache Primes
Paulo Ribenboim: The New Book of Prime Number Records: Additions and Errata
Warut Roonguthai: Proth primes and Cunningham Chains
N. J. A. Sloane: On-Line Encyclopedia of Integer Sequences
W. F. C. Taylor: A Tale of Two Conjectures
Eric W. Weisstein: Prime Quadruplet
Eric W. Weisstein: Prime k-Tuples Conjecture
Eric W. Weisstein: Prime Constellation
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 > 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
H C ∫u = 2 to x du / (log u)k,
where
H = ∏p ≤ k pk − 1 (p − v) / (p − 1)k ∏p > k, p|D (p − v) / (p − k),
C = ∏p > k p(k − 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 of the bs.
The first product in H is over the primes not greater than k, the second is over the primes greater than k which divide D and the product C 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, H = 2, and C = C2, the twin prime constant given above.
[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.
[F96f] Tony Forbes, Prime k-tuplets, NMBRTHRY Mailing List, December 1996.
[F97f] Tony Forbes, Prime 17-tuplet, NMBRTHRY Mailing List, September 1997.
[F02] Tony Forbes, Titanic prime quintuplets, M500 189 (December, 2002), 12-13.
[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.