Prime k-tuplets

Abstract

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.

The intended audience is anyone with an interest in mathematics, especially prime numbers.

Prepared by Tony Forbes.

Contents

  1. Introduction
  2. The Largest Known Prime Twins
  3. The Largest Known Prime Triplets
  4. The Largest Known Prime Quadruplets
  5. The Largest Known Prime Quintuplets
  6. The Largest Known Prime Sextuplets
  7. The Largest Known Prime Septuplets
  8. The Largest Known Prime Octuplets
  9. The Largest Known Prime 9-tuplets
  10. The Largest Known Prime 10-tuplets
  11. The Largest Known Prime 11-tuplets
  12. The Largest Known Prime 12-tuplets
  13. The Largest Known Prime 13-tuplets
  14. The Largest Known Prime 14-tuplets
  15. The Largest Known Prime 15-tuplets
  16. The Largest Known Prime 16-tuplets
  17. The Largest Known Prime 17-tuplets
  18. The Largest Known Prime 18-tuplets
  19. Odds and Ends
  20. Links to Related Material
  21. Mathematical Background
  22. References
  23. About This Site

1. Introduction

Prime Numbers

Prime numbers are the building blocks of arithmetic. They are a special type of number because they cannot be broken down into smaller factors. 13 is prime because 13 is 1 times 13 (or 13 times 1), and that's it. There's no other way of expressing 13 as something times something. On the other hand, 12 is not prime because it splits into 2 times 6, or 3 times 4.

The first prime is 2. The next is 3. Then 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503 and so on. If you look at the first 10000 primes, you will see a list of numbers with no obvious pattern. There is even an air of mystery about them; if you didn't know they were prime numbers, you would probably have no idea how to continue the sequence. Indeed, if you do manage discover a simple pattern, you will have succeeded where some of the finest brains of all time have failed. For this is an area where mathematicians are well and truly baffled.

We do know fair amount about prime numbers, and an excellent starting point if you want to learn more about the subject is Chris Caldwell's web site: The Largest Known Primes. We know that the sequence of primes goes on for ever. We know that it thins out. The further you go, the rarer they get. We even have a simple formula for estimating roughly how many primes there are up to some large number without having to count them one by one. However, even though prime numbers have been the object of intense study by mathematicians for hundreds of years, there are still fairly basic questions which remain unanswered.

Prime Twins

If you look down the list of primes, you will quite often see two consecutive odd numbers, like 3 and 5, 5 and 7, 11 and 13, 17 and 19, or 29 and 31. We call these pairs of prime numbers {p, p+2} prime twins.

The evidence suggests that, however far along the list of primes you care to look, you will always eventually find more examples of twins. Nevertheless, - and this may come as a surprise to you - it is not known whether this is in fact true. Possibly they come to an end. But it seems more likely that - like the primes - the sequence of prime twins goes on forever. However, Mathematics has yet to provide a rigorous proof.

One of the things mathematicians do when they don't understand something is produce bigger and better examples of the objects that are puzzling them. We run out of ideas, so we gather more data - and this is just what we are doing at this site; if you look ahead to section 2, you will see that I have collected together the ten largest known prime twins.

Prime Triplets

If you search the list for triples of primes {p, p+2, p+4}, you will not find very many. In fact there is only one, {3, 5, 7}, right at the beginning. And it's easy to see why. 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.

Prime Quadruplets

Similar considerations apply to groups of four, where this time we require each of {p, p+2, p+6, p+8} to be prime. Once again, it looks as if they go on indefinitely. The smallest is {5, 7, 11, 13}. We don't count {2, 3, 5, 7} even though it is a denser grouping. It is an isolated example which doesn't fit into the scheme of things. Nor, for more technical reasons, do we count {3, 5, 7, 11}.

The sequence continues with {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, .... The usual name is prime quadruplets, although I have also seen the terms full house, inter-decal prime quartet (!) and prime decade - a reference to the pattern made by their decimal digits [M96a, M96b]. 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.

Prime k-tuplets

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.

Notation

The symbol for multiplication is an asterisk: x*y means x times y. The symbol "^" means "to the power of": Thus x^2 = x*x, x^3 = x*x*x and so on.

For k > 2, the abbreviation N + b1, b2, ..., bk denotes the k numbers {N + b1, N + b2, ..., N + bk}. Prime twins are represented by N +/- 1, which is short for N plus one and N minus one.

I also use the notation p# of Caldwell and Dubner [CD93] as a convenient shorthand for 2*3*5*...*p, the product of all the primes up to and including p.

Finally,

I would like to keep this site as up to date as possible. Therefore, can I urge you to please send any new, large prime 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 300 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: tonyforbes@ltkz.demon.co.uk.

2. The Largest Known Prime Twins

2003663613*2^195000 +/- 1 (58711 digits, Jan 2007, Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Michaek Kwok, Andrea Pacini, Rytis Slatkevicius)

100314512544015*2^171960 +/-1 (51780 digits, Dec 2006, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)

16869987339975 * 2^171960 +/- 1 (51779 digits, Sep 2005, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)

33218925 * 2^169690 +/- 1 (51090 digits, 2002, Daniel Papp & Yves Gallot)

60194061 * 2^114689 +/- 1 (34533 digits, 2002, David Underbakke)

1765199373 * 2^ 107520 +/- 1 (32376 digits, 2002, James McElhatton & Yves Gallot)

318032361 * 2^107001 +/- 1 (32220 digits, 2001, David Underbakke & Phil Carmody)

1807318575 * 2^98305 +/- 1 (29603 digits, 2001, David Underbakke & Phil Carmody)

7473214125 * 2^83125 +/- 1 (25033 digits, 2006, David Underbakke)

11694962547 * 2^83124 +/- 1 (25033 digits, 2006, David Underbakke)

See Chris Caldwell, The Largest Known Primes, for further (and more up-to-date) information.

3. The Largest Known Prime Triplets

(63140956174*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35 + 1, 5, 7 (5132 digits, Oct 2005, Ken Davis)

(63095588824*205881*4001#*(205881*4001#+1)+210)*(205881*4001#-1)/35 + 7, 11, 13 (5132 digits, Oct 2005, Ken Davis)

(62258488321368*3331#*(1037*3331#+1)+210)*(1037*3331#-1)/35 + 5, + 7, + 11 (4259 digits, Jul 2003, David Broadhurst)

(22877907949788*3331#*(1037*3331#+1)+210)*(1037*3331#-1)/35 + 5, + 7, + 11 (4259 digits, Jul 2003, David Broadhurst)

(108748629354*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, + 13 (4135 digits, Sep 2002, David Broadhurst, PFGW)

(90159302514*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5, + 7, + 11 (4135 digits, Sep 2002, David Broadhurst, PFGW)

(39553075974*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 5, + 7, + 11 (4135 digits, Sep 2002, David Broadhurst, PFGW)

(18599651274*4436*3251#*(4436*3251#+1)+210)*(4436*3251#-1)/35 + 7, + 11, +13 (4134 digits, Sep 2002, David Broadhurst, PFGW)

(61504372896*5119*3163#*(5119*3163#+1)+210)*(5119*3163#-1)/35 + 5, + 7, + 11 (4019 digits, Aug 2002, David Broadhurst, PFGW)

813529138*1939938^500 -1, +1, +5 (3153 digits, Jul 2002, Norman Luhn, NewPgen, PFGW, PRIMO)

More

4. The Largest Known Prime Quadruplets

4104082046*4800# + 5651 + 0, 2, 6, 8 (2058 digits, Apr 2005, Norman Luhn, Primo)

11024895887*3500# + 855731 +0, 2, 6, 8 (1491 digits, Feb 2003, Norman Luhn, Primo)

10271674954*2999# + 3461 + 0, 2, 6, 8 (1284 digits, Feb 2002, Michael Bell, Michael Davison, Matt Jack, Ronald Lau, Graeme Leese and Ben Lowing)

11^1101+45917626999140 + 0, 2, 6, 8 (1147 digits, Sep 2005, Ronny Edler)

11^1101+34264768249680 + 0, 2, 6, 8 (1147 digits, Sep 2005, Ronny Edler)

700209251206546*3^2239 -1, +1, +5, +7 (1084 digits, Aug 2005, Michael Gillion & George Woltman)

18973472837*2503#/35 + 7, +5, +1, -1 (1070 digits, Aug 2005, Gary Chaffey, PRIMO)

283534892623*2500# + 1091261 + 0, 2, 6, 8 (1068 digits, Apr 2006, Norman Luhn)

109267227191*2500# + 10531091 + 0, 2, 6, 8 (1068 digits, Jul 2001, Norman Luhn, PFGW, PRIMO)

8954571083387140525*(2^3423 - 2^1141) - 6*2^1141 - 7, -5, -1, +1 (1050 digits, Oct 1999, Tony Forbes)

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5. The Largest Known Prime Quintuplets

283534892623*2500# + 1091261 + 0, 2, 6, 8, 12 (1069 digits, Apr 2006, Norman Luhn)

31969211688*2400#+16061 + 0, 2, 6, 8, 12 (1034 digits, Jul 2002, Norman Luhn [F02], APSieve, PFGW, PRIMO)

912143859*1223# + 463001711 + 0, 2, 6, 8, 12  (522 digits, Mar 2004, Donovan Johnson)

19685846183*1200# + 6005891 + 0, 2, 6, 8, 12 (511, digits, May 2002, Norman Luhn, PFGW, PRIMO)

14519751105*1050# + 1042090781 + 0, 2, 6, 8, 12 (450 digits, Apr 2002, Michael Hannigan)

338769039776*1000#+16061 +0, 2, 6, 8, 12 (427 digits, Jan 2006, Norman Luhn)

328481121285*1000# + 16057 + 4, 6, 10, 12, 16 (427 digits, Jan 2006, Norman Luhn)

328481121285*1000# + 16057 +0, 4, 6, 10, 12 (427 digits, Jan 2006, Norman Luhn)

319335512503*1000#+16061 +0, 2, 6, 8, 12 (427 digits, Jan 2006, Norman Luhn)

305727029371*1000#+16061 +0, 2, 6, 8, 12 (427 digits, Jan 2006, Norman Luhn)

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6. The Largest Known Prime Sextuplets

328481121285*1000# + 16057 +0, 4, 6, 10, 12, 16 (427 digits, Jan 2006, Norman Luhn)

138765468778 * 850# + 2822707 + 0, 4, 6, 10, 12, 16 (362 digits, Apr 2004, Norman Luhn)

8398544501*710# + 2000472907 + 0, 4, 6, 10, 12, 16 (306 digits, Aug 2003, Torbjörn Alm & Jens Kruse Andersen, VFYPR)

110282080125*700# + 6005887 + 0, 4, 6, 10, 12, 16 (301 digits, Oct 2001, Norman Luhn, PRIMO)

97953153175*670# + 16057 + 0, 4, 6, 10, 12, 16 (290 digits, Apr 2001, Michael Bell, Graeme Leese, Michael Davison, APSieve and Titanix)

86450022463*570# + 1000000587445747 + 0, 4, 6, 10, 12, 16 (242 digits, Jan 2001, Norman Luhn)

1189609319*503#/613777 + 446215867 +0, +4, +6, +10, +12, +16 (213 digits, Sep 2000, Michael Bell)

82248305245 * 43# * 2^479 + 16057 + 0, 4, 6, 10, 12, 16 (172 digits, 1997, A.O.L. Atkin)

10^160 + 72849172960797 + 0, 4, 6, 10, 12, 16 (161 digits, Oct 2005, Richard Miller)

2^512 + 6638977280721 + 0, 4, 6, 10, 12, 16 (155 digits, 1996, Tony Forbes [F96f])

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7. The Largest Known Prime Septuplets

251733155478*650#+1146779 + 0, 2, 8, 12, 14, 18, 20 (282 digits, Jan 2006, Norman Luhn)

76794640264*509# + 5132201 + 0, 2, 6, 8, 12, 18, 20 (223 digits, Sep 2004, Jens Kruse Andersen)

1839198074074 * 500# + 165701 + 0, 2, 6, 8, 12, 18, 20 (219 digits, Jun 2004, Norman Luhn)

15350892425771 * 465# + 88799 +0, 2, 8, 12, 14, 18, 20 (206 digits, Aug 2005, Norman Luhn)

12874261020824 * 465# + 88793 +6, 8, 14, 18, 20, 24, 26 (206 digits, Aug 2005, Norman Luhn)

11435332301154 * 465# + 88799 +0, 2, 8, 12, 14, 18, 20 (206 digits, Aug 2005, Norman Luhn)

10556596219791 * 465# + 88799 +0, 2, 8, 12, 14, 18, 20 (206 digits, Aug 2005, Norman Luhn)

9379898589313 * 465# + 88799 +0, 2, 8, 12, 14, 18, 20 (206 digits, Aug 2005, Norman Luhn)

8380138890134 * 465# + 88799 +0, 2, 8, 12, 14, 18, 20 (206 digits, Aug 2005, Norman Luhn)

5585090256052 * 465# + 88799 +0, 2, 8, 12, 14, 18, 20 (206 digits, Aug 2005, Norman Luhn)

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8. The Largest Known Prime Octuplets

12874261020824 * 465# + 88793 +0, 6, 8, 14, 18, 20, 24, 26 (206 digits, Aug 2005, Norman Luhn)

97510235*421# + 322355908991 + 0, 2, 6, 8, 12, 18, 20, 26 (180 digits, Jan 2005, Torbjörn Alm & Jens Kruse Andersen)

4319152256906 * 400# + 1277 + 0, 2, 6, 12, 14, 20, 24, 26 (174 digits, Nov 2004, Norman Luhn)

65677369861*380# + 18000020393471 + 0, 2, 6, 8, 12, 18, 20, 26 (165 digits, Aug 2003, Torbjörn Alm & Jens Kruse Andersen)

15234072433401 * 375# + 43813839521 + 0, 2, 6, 8, 12, 18, 20, 26 (164 digits, May 2002, Norman Luhn)

243551752728*320# + 1277 + 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)

118195793431939*300# + 103498931 + 0, 2, 6, 8, 12, 18, 20, 26 (135 digits, Feb 2005, Norman Luhn)

98481667857555*300# + 103498931 + 0, 2, 6, 8, 12, 18, 20, 26 (135 digits, Feb 2005, Norman Luhn)

90421624808713*300# + 103498931 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (135 digits, Feb 2005, Norman Luhn)

83957036111044*300# + 103498931 + 0, 2, 6, 8, 12, 18, 20, 26 (135 digits, Feb 2005, Norman Luhn)

78982397556852*300# + 103498931 + 0, 2, 6, 8, 12, 18, 20, 26 (135 digits, Feb 2005, Norman Luhn)

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9. The Largest Known Prime Nonuplets

90421624808713*300# + 103498931 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (135 digits, Feb 2005, Norman Luhn)

1619062142 * 255# + 53344165991 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (111 digits, Aug 2003, Thomas Wolter & Jens Kruse Andersen)

388793398651*250# + 1042090781 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (110 digits, Feb 2001, Norman Luhn)

2242445342405*230# + 39058751 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Jun 2003, Norman Luhn)

24698258*239# + 28606476153371 + 6, 8, 12, 18, 20, 26, 30, 32, 36 (104 digits, Aug 2004, Norman Luhn & Jens Kruse Andersen)

24698258*239# + 28606476153371 + 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, Aug 2004, Jens Kruse Andersen)

24698258*239# + 28606476153371 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Aug 2004, Jens Kruse Andersen)

18188893*239# + 7597110015611 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Aug 2004, Jens Kruse Andersen)

12023698*239# + 19623646397471 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Aug 2004, Jens Kruse Andersen)

9808759*239# + 9127517704421 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (104 digits, Aug 2004, Jens Kruse Andersen)

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10. The Largest Known Prime Decuplets

24698258*239# + 28606476153371 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, Aug 2004, Jens Kruse Andersen)

72613488698235 * 227# + 39058751 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (103 digits, Apr 2004, Norman Luhn)

36273553*157# + 106263743005151 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)

35405054*157# + 143751012544871 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)

35078052*157# + 398861548425071 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)

34101658*157# + 164826429367331 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)

33744069*157# + 243858308984021 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)

33395983*157# + 49822093470881 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)

31812413*157# + 394317153630131 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)

30972388*157# + 218344297501061 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)

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11. The Largest Known Prime 11-tuplets

24698258*239# + 28606476153371 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (104 digits, Aug 2004, Norman Luhn & Jens Kruse Andersen)

35078052*157# + 398861548425071 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (70 digits, Feb 2004, Jens Kruse Andersen & Hans Rosenthal)

34101658*157# + 164826429367331 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (70 digits, Feb 2004, Hans Rosenthal & Jens Kruse Andersen)

92119245478633*130# + 21816911 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (63 digits, Dec 2003, Norman Luhn)

58187756*110# + 2320048690691 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (53 digits, Aug 2003, Jens Kruse Andersen)

11450665899501*101# + 39058751 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (52 digits, Aug 2003, Norman Luhn)

8747677*107# + 4008289033283651 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)

8519951*107# + 4108713619105211 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)

8486221*107# + 4549290807806861 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)

7862516*107# + 4453056384461801 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)

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12. The Largest Known Prime dodecuplets

8486221*107# + 4549290807806861 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, May 2006, Dirk Augustin & Jens Kruse Andersen)

434161298*89# + 612442658382671 + 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 + 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 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (43 digits, Jul 2003, Jens Kruse Andersen)

5027317106963 * 75# + 1418575498567 + 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (42 digits, Nov 2001, Norman Luhn)

720345861287087*70# + 8393501 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)

564115572162757*70# + 8393501 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)

412477355651067*70# + 8393501 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)

186519833921143*70# + 8393501 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)

130171389197721*70# + 8393501 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (40 digits, Aug 2003, Norman Luhn)

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13. The Largest Known Prime 13-tuplets

26697593*67# + 315911634133211 + 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 + 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 + 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 + 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 + 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 + 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 + 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 + 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)

22865416*67# + 58526097915401 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (33 digits, Feb 2005, Christ van Willegen & Jens Kruse Andersen)

22516115*67# + 73934561215633 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (33 digits, Jan 2005, Christ van Willegen & Jens Kruse Andersen)

More

14. The Largest Known Prime 14-tuplets

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

99999999948164978600250563546400 + 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 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)

1124495649695212998778414534811 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Aug 2003, Thomas Wolter & Jens Kruse Andersen)

1100916249233879857334075234831 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)

1000000008282508019026959814211 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Dec 2000, Joerg Waldvogel & Peter Leikauf)

1000000007541367760266886291861 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Dec 2000, Joerg Waldvogel & Peter Leikauf)

1000000006672161724368529625351 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Dec 2000, Joerg Waldvogel & Peter Leikauf)

1000000005832631360266813468481 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Dec 2000, Joerg Waldvogel & Peter Leikauf)

1000000005644941246959007679801 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (31 digits, Dec 2000, Joerg Waldvogel & Peter Leikauf)

More

15. The Largest Known Prime 15-tuplets

99999999948164978600250563546400 + 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 + 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 + 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 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (30 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)

10252256693298561414756287 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (26 digits, Oct 2004, Jens Kruse Andersen)

2845372542509911868266817 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (25 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)

1906230835046648293290047 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (25 digits, Feb 2001, Joerg Waldvogel & Peter Leikauf)

163027495131423420474917 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (24 digits, May 2001, Peter Leikauf & Joerg Waldvogel)

123452114023762529883167 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (24 digits, Jul 1999, Joerg Waldvogel)

110885131130067570042707 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (24 digits, May 2001, Peter Leikauf & Joerg Waldvogel)

More

16. The Largest Known Prime 16-tuplets

10252256693298561414756287 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (26 digits, Oct 2004, Jens Kruse Andersen)

2845372542509911868266817 + 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 + 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 + 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 + 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 + 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 + 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 + 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 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (23 digits, Mar 1999, Tony Forbes)

3259125690557440336637 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (22 digits, Sep 1997, Tony Forbes [F97f])

More

17. The Largest Known Prime 17-tuplets

2845372542509911868266811 + 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 + 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 + 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 + 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 + 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 + 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 + 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 + 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 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (23 digits, Aug 1998, Tony Forbes)

47624415490498763963983 + 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)

18. The Largest Known Prime 18-tuplets

2845372542509911868266807 + 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 + 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

19. Odds and Ends

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]

Site History

20. Links to related material

Chris K. Caldwell: The Largest Known Primes

Chris K. Caldwell: Paulo Ribenboim, The New Book of Prime Number Records: Additions and Errata

Chris K. Caldwell: All known twin primes with at least 1000 digits

Jens Kruse Andersen: Consecutive Primes in Arithmetic Progression

Y. H. Huen: Goldbach Sequences

Dr. Minh. L. Perez Press: Smarandache Primes

Warut Roonguthai: Proth primes and Cunningham Chains

Dirk Augustin: Cunningham Chain records

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

21. Mathematical Background

Definition

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, and 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.

Patterns of Prime k-tuplets

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.

Primality Checking

In keeping with similar published lists, I have decided not to accept anything other than true, verifiable 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 (APRT-CLE in UBASIC, for example), or Atkin and Morain's Elliptic Curve program, ECPP.

Primes

Euclid proved that there are infinitely many primes. Paulo Ribenboim [Rib95] has collected together a considerable number of different proofs of this important theorem. My favourite (which is not in Ribenboim's book) goes like this: We have

product{p prime, 1/(1 - 1 / p^2)} = sum{n=1 to infinity, 1 / n^2} = pi^2/6.

But pi^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:

integral{u=0 to x, du/(log u)} + error term,

where the error term is bounded above by A x exp(-B sqrt(log x)) for some constants A and B. With more work (H.-E. Richert, 1967) the exponent 1/2 of log x in this last expression can be replaced by 3/5. 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 the function A sqrt(x) log x.

The Twin Prime Conjecture

G.H. Hardy & J.E. Littlewood did the first serious work on the distribution of prime twins. In their paper 'Some problems of Partitio Numerorum: III...' [HL22], they conjectured a formula for the number of twins between 1 and x:

2 C2 x / (log x)^2,

where C2 = product{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 sum{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 Hardy-Littlewood Prime k-tuple Conjecture

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 integral{from u=2 to x, du / (log u)^k},

where

H = product{p <= k, p^(k-1) (p-v) / (p-1)^k}* product{p>k, p|D, (p-v) / (p-k)},

C = product{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 b's.

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.

22. References

[BLS75] John Brillhart, D.H. Lehmer & J.L. Selfridge, New primality criteria and factorizations of 2^m +/- 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.

[F96a] Tony Forbes, A small collection of prime k-tuplets, NMBRTHRY Mailing List, January 1996.

[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.

[M96a] Mike Mudge, Pounding the Beat, Personal Computer World March 1996, 309.

[M96b] Mike Mudge, Going Back to Your Roots, Personal Computer World December, 1996, 285.

[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.

23. About This Site

Classification: SCIENCE / MATHEMATICS / NUMBERS / PRIME NUMBERS / DISTRIBUTION OF PRIME NUMBERS.

Intended Audience: Anyone with an interest in mathematics, especially prime numbers. I have tried to keep the mathematical sophistication of at least the introductory section to a minimal level. Comments and suggestions are welcome. Also new links and, of course, new additions to the lists.

Prepared by: Tony Forbes. Email address: tonyforbes @ltkz. demon. co. uk..

Last updated: 16 January 2007.