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.

- Introduction
- The Largest Known Prime Twins
- The Largest Known Prime Triplets
- The Largest Known Prime Quadruplets
- The Largest Known Prime Quintuplets
- The Largest Known Prime Sextuplets
- The Largest Known Prime Septuplets
- The Largest Known Prime Octuplets
- The Largest Known Prime 9-tuplets
- The Largest Known Prime 10-tuplets
- The Largest Known Prime 11-tuplets
- The Largest Known Prime 12-tuplets
- The Largest Known Prime 13-tuplets
- The Largest Known Prime 14-tuplets
- The Largest Known Prime 15-tuplets
- The Largest Known Prime 16-tuplets
- The Largest Known Prime 17-tuplets
- The Largest Known Prime 18-tuplets
- Odds and Ends
- Links to Related Material
- Mathematical Background
- References
- About This Site

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 [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.

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.

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* + *b*1,
*b*2, ..., *b**k* denotes the *k* numbers {*N* +
*b*1,
*N* + *b*2, ...,
*N* + *b**k*}. 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*.

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.

**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.

**(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)

**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)

**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)

**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])

**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)

**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)

**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)

**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)

**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)

**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)

**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)

**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)

**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)

**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])

**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)

**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

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]

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

A **prime k-tuplet** is a sequence of

More precisely: We first define ** s(k)** to be the
smallest number

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, 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.

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*, d*u*/(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*.

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

2 *C*2 *x* / (log *x*)^2,

where *C*2 = 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 *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* *b*1,
*b*2, ..., *b**k* *be k distinct integers. Then the number
of groups of primes* *N* + *b*1,
*N* + *b*2, ..., *N* + *bk* *between* 2 *and x
is approximately*

*H* *C* integral{from *u*=2 to
*x*, d*u* / (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*
*b*1, *b*2, ..., *b**k 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, *b*1 = 0 and *b*2 = 2, then
*v*(2) = 1, *v*(*p*) = *p* - 1 for *p* > 2,
*H* = 2, and *C* = *C*2, the twin prime constant given above.

[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.

**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.