At this site I have collected together all the largest known examples of certain
types of dense clusters of prime numbers. The idea is to generalise the notion
of *prime twins* - pairs of prime numbers {*p*,
*p* + 2} - to groups of three or more.

Prepared by Tony Forbes; anthony.d.forbes@gmail.com.

Site address: http://anthony.d.forbes.googlepages.com/ktuplets.htm.

16 Jun 2019

Prime 7-tuplet (NEW WORLD RECORD!)

32821868878860201045633341031688415601401701228
32878265333984717524446848642006351778066196724473
92249620201536539259942023218972369026762290403609
01005487309186655777663859063397693729163631275766
07799875309038457637116938538279395260265064447747
74261236889041020217108597484837589978261046949778
71991825164994665583879769659044973939714534960362
41885200541893611077817261813672809971503287259089 * 317# + 1068701 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (527 digits, 16 Jun 2019, Vidar Nakling, RIEMINER0.9, PRIMO)

29 May 2019

Prime 11-tuplet (NEW WORLD RECORD!)

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 * 229# + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (107 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

Prime 10-tuplets

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 9268026349694711 * 229# + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4265997347677925 * 229# + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (107 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

Smallest 400-digit prime sextuplet

10^399 + 33756090918084087 + *d*, *d* = 0, 4, 6, 10, 12, 16 (400 digits, 7 May 2019, Norman Luhn, PRIMO)

Smallest titanic prime quintuplet to pattern d = 0, 4, 6, 10, 12

10^999 + 3818999670116007 + *d*, *d* = 0, 4, 6, 10, 12 (1000 digits, 9 May 2019, Norman Luhn, PRIMO)

Prime triplets

647935598824239 * 2^{33619} + *d*, *d* = −1, 1, 5 (10136 digits, 22 May 2019, Peter Kaiser, PRIMO)

209102639346537 * 2^{33620} + *d*, *d* = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

185353103135997 * 2^{33620} + *d*, *d* = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

162615027598677 * 2^{33620} + *d*, *d* = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

4 May 2019

Prime 20-tuplet

686962597479437604159786541481 + *d*, *d* = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, April 27, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

24 April 2019

Prime triplet (NEW WORLD RECORD!)

4111286921397 * 2^{66420} + *d*, *d* = −1, 1, 5 (20008 digits, 24 Apr 2019, Peter Kaiser, POLYSIEVE, LLR, PRIMO)

9 April 2019

Prime 7-tuplet (the smallest with 320 digits)

10^319 + 2219844666811981651 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (320 digits, 9 Apr 2019, Norman Luhn, PRIMO)

7 April 2019

Prime 20-tuplet (NEW WORLD RECORD!)

701870455949526598513130862539 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, April 7, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

27 February 2019

Prime quadruplet (NEW WORLD RECORD! And first to exceed 10000 digits!)

667674063382677 * 2^{33608} + *d*, *d* = −1, 1, 5, 7 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)

23 February 2019

Prime 20-tuplet, (NEW WORLD RECORD!)

667424014858149638371951648871 + *d*, *d* = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, February 18, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

1 January 2019

Prime 21-tuplet, (NEW WORLD RECORD! Includes new world record prime 19-tuplet)

622803914376064301858782434517 + *d*, *d* = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84 (30 digits, December 27, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

26 December 2018

Prime 20-tuplets, (including a new world record and the first large one)

639121700726230052098229452019 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, December 23, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

594750459626903777773717631519 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, December 23, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

593820854957327357933627374349 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, December 23, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

3941119827895253385301920029 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (28 digits, June 24, 2014, Raanan Chermoni & Jaroslaw Wroblewski)

Prime 15-tuplets (first of each pattern with 25 digits)

1000543345438817590469987 + *d*, *d* = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (25 digits, December 2018, Norman Luhn)

1000543338893999053267943 + *d*, *d* = 0, 6, 8, 14, 20, 24, 26, 30, 36, 38, 44, 48, 50, 54, 56 (25 digits, December 2018, Norman Luhn)

1000246552183249816179851 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (25 digits, December 2018, Norman Luhn)

1000009162985306844349997 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (25 digits, December 2018, Norman Luhn)

- 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
- The Largest Known Prime 19-tuplets
- The Largest Known Prime 20-tuplets
- The Largest Known Prime 21-tuplets
- Summary
- Odds and Ends
- Links to Related Material
- Mathematical Background
- References

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

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

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

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

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

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

If you search the list for triples of primes {*p*, *p* + 2, *p* + 4},
you will not find very many. In fact there is only one, {3, 5, 7}, right at the
beginning. And it's easy to see why. As G. H. Hardy & J. E. Littlewood
observed [HL22], at least one of the three is divisible by 3.

Obviously it is asking too much to squeeze three primes into a range of
four. However, if we increase the range to six and look for combinations {*p*,
*p* + 2, *p* + 6} or {*p*,
*p* + 4, *p* + 6}, we find plenty of examples, beginning with {5, 7,
11}, {7, 11, 13}, {11, 13, 17}, {13, 17, 19}, {17, 19, 23}, {37, 41, 43}, ....
These are what we call *prime triplets*, and one of the main objectives of
this site is to collect together all the largest known examples. Just as with
twins, it is believed - but not known for sure - that the sequence of prime
triplets goes on for ever.

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

The sequence continues with {11, 13, 17, 19}, {101, 103, 107, 109}, {191,
193, 197, 199}, {821, 823, 827, 829}, .... The usual name is *prime
quadruplets*, although I have also seen the terms *full house*, *inter-decal
prime quartet* (!) and *prime decade* - a reference to the pattern made
by their decimal digits. All primes greater than 5 end in one of 1,
3, 7 or 9, and the four primes in a (large) quadruplet always occur in the same
ten-block. Hence there must be exactly one with each of these unit digits. And
just to illustrate the point, here is another example; the smallest prime
quadruplet of 50 digits, found by G. John Stevens in 1995 [S95]:

10000000000000000000000000000000000000000058537891,

10000000000000000000000000000000000000000058537893,

10000000000000000000000000000000000000000058537897,

10000000000000000000000000000000000000000058537899.

We can go on to define *prime quintuplets*, *sextuplets*,
*septuplets*, *octuplets*, *nonuplets*, and so on. I had to go
to the full *Oxford English Dictionary* for the last one - the *Concise
Oxford* jumps from 'octuplets' to 'decuplets'. The *OED* also defines
'dodecuplets', but apparently there are no words for any of the others.
Presumably I could make them up, but instead I shall use the number itself when
I want to refer to, for example, prime 11-tuplets. I couldn't find the general
term '*k*-tuplets' in the *OED* either, but it is the word that
seems to be in common use by the mathematical community.

For now, I will define a **prime k-tuplet** as a sequence of
consecutive prime numbers such that the distance between the first and the last
is in some sense as small as possible. If you think I am being too vague, there
is a more precise
definition later on.

At this site I have collected together what I believe to be the largest
known prime *k*-tuplets for *k* = 2, 3, 4, ..., 20 and 21. I do not
know of any prime *k*-tuplets for
*k* greater than 21, except for the ones that occur near the beginning of
the prime number sequence.

Multiplication is often denoted by an asterisk:
*x***y* is *x* times *y*.

For *k* > 2, the somewhat bizarre notation *N* + *b*_{1},
*b*_{2}, ..., *b*_{k} is used (only in linked pages) to denote the *k* numbers {*N* +
*b*_{1},
*N* + *b*_{2}, ...,
*N* + *b*_{k}}.

Prime twins are represented as
*N* ± 1, which is short for *N* plus one and *N* minus one.

I also use the notation *n*# of Caldwell and Dubner [CD93] as a
convenient shorthand for the product of all the primes less than or
equal to *n*. Thus, for example, 20# = 2*3*5*7*11*13*17*19 = 9699690.

I would like to keep this site as up to date as possible. Therefore, can I
urge you to *please send any new, large prime k-tuplets to me*. You can
see what I mean by 'large' by studying the lists. If the numbers are not too
big, say up to 500 digits, I am willing to double-check them myself. Otherwise I
would appreciate some indication of how you proved that your numbers are true
primes. Email address: anthony.d.forbes@gmail.com.

**2996863034895 * 2 ^{1290000} ± 1** (388342 digits, Sep 2016, Tom Greer, TWINGEN, PRIMEGRID, LLR)

3756801695685 * 2^{666669} ± 1 (200700 digits, Dec 2011, Timothy Winslow, TWINGEN, PRIMEGRID, LLR)

65516468355 * 2^{333333} ± 1 (100355 digits, Aug 2009, Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR)

12770275971 * 2^{222225} ± 1 (66907 digits, Jul 2017, Bo Tornberg, TWINGEN, LLR TWIN)

70965694293 * 2^{200003} ± 1 (60219 digits, Apr 2016, S. Urushihata)

66444866235 * 2^{200003} ± 1 (60218 digits, Apr 2016, S. Urushihata)

4884940623 * 2^{198800} ± 1 (59855 digits, Jul 2015, Kwok, PSIEVE, LLR)

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

38529154785 * 2^{173250} ± 1 (52165 digits, Jul 2014, Serge Batalov, NEWPGEN, LLR)

194772106074315 * 2^{171960} ± 1 (51780 digits, Jun 2007, Antal Járai, Gabor Farkas, Timea Csajbok & János Kasza)

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

**4111286921397 * 2 ^{66420} + d, d = −1, 1, 5** (20008 digits, 24 Apr 2019, Peter Kaiser, POLYSIEVE, LLR, PRIMO)

6521953289619 * 2^{55555} + *d*, *d* = −5, −1, 1 (16737 digits, Apr 2013, Peter Kaiser)

3221449497221499 * 2^{34567} + *d*, *d* = −1, 1, 5 (10422 digits, Sep 2015, Peter Kaiser, NEWGEN, LLR, PRIMO5)

1288726869465789 * 2^{34567} + *d*, *d* = −5, −1, +1 (10421 digits, Apr 2014, Peter Kaiser)

647935598824239 * 2^{33619} + *d*, *d* = −1, 1, 5 (10136 digits, 22 May 2019, Peter Kaiser, PRIMO)

209102639346537 * 2^{33620} + *d*, *d* = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

185353103135997 * 2^{33620} + *d*, *d* = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

162615027598677 * 2^{33620} + *d*, *d* = −1, 1, 5 (10135 digits, 22 May 2019, Peter Kaiser, PRIMO)

667674063382677 * 2^{33608} + *d*, *d* = 1, 5, 7 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)

667674063382677 * 2^{33608} + *d*, *d* = −1, 1, 5 (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)

**667674063382677 * 2 ^{33608} + d, d = −1, 1, 5, 7** (10132 digits, 27 Feb 2019, Peter Kaiser, PRIMO)

4122429552750669 * 2^{16567} + *d*, *d* = −1, 1, 5, 7 (5003 digits, Mar 2016, Peter Kaiser, GSIEVE, NewPGen, LLR, PRIMO)

2673092556681 * 15^{3048} + *d*, *d* = −4, −2, 2, 4 (3598 digits, Sep 2015, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

2339662057597 * 10^{3490} + *d*, *d* = 1, 3, 7, 9 (3503 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

305136484659 * 2^{11399} + *d*, *d* = −1, 1, 5, 7 (3443 digits, Sep 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

722047383902589 * 2^{11111} + *d*, *d* = −1, 1, 5, 7 (3360 digits, Apr 2013, Reto Keiser, NEWPGEN, PFGW, PRIMO)

43697976428649 * 2^{9999} + *d*, *d* = −1, 1, 5, 7 (3024 digits, Mar 2012, Peter Kaiser)

46359065729523 * 2^{8258} + *d*, *d* = −1, 1, 5, 7 (2500 digits, Nov 2011, Reto Keiser, NEWPGEN, PFGW, PRIMO)

1367848532291 * 5591# / 35 + *d*, *d* = −1, 1, 5, 7 (2401 digits, Aug 2011, Norman Luhn, NEWPGEN, PFGW, PRIMO)

25796119248 * 4987# / 35 + *d*, *d* = −1, 1, 5, 7 (2135 digits, May 2011, Gary Chaffey)

**394254311495 * 3733# / 2 + d, d = -8, -4, -2, 2, 4** (1606 digits, Nov 2017, Serge Batalov, NEWPGEN, OPENPFGW, PRIMO)

2316765173284 * 3600# + 16061 + *d*, *d* = 0, 2, 6, 8, 12 (1543 digits, 16 Oct 2016, Norman Luhn, PRIMO)

163252711105 * 3371# / 2 + *d*, *d* = −8, −4, −2, 2, 4 (1443 digits, Jan 2014, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

9039840848561 * 3299# / 35 + *d*, *d* = −5, −1, 1, 5, 7 (1401 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

699549860111847 * 2^{4244} + *d*, *d* = −1, 1, 5, 7, 11 (1293 digits, Dec 2013, Reto Keiser, R. Gerbicz, PFGW, PRIMO)

566650659276 * 2621# + 1615841 + *d*, *d* = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)

554729409262 * 2621# + 1615841 + *d*, *d* = 0, 2, 6, 8, 12 (1117 digits, Dec 2013, David Broadhurst, PRIMO, OpenPFGW)

424232794973 * 2600# + 43777 + *d*, *d* = 0, 4, 6, 10, 12 (1107 digits, Mar 2009, Norman Luhn, PRIMO)

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

96972480423104 * 2400# + 19417 + *d*, *d* = 0, 4, 6, 10, 12 (1038 digits, Nov 2012, Norman Luhn, PRIMO)

**28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16** (1037 digits, 14 Mar 2016, Norman Luhn, APSIEVE, PRIMO)

6646873760397777881866826327962099685830865900246688640856 * 1699# + 43777 + *d*, *d* = 0, 4, 6, 10, 12, 16 (780 digits, 8 Nov 2018, Vidar Nakling, PRIMO)

29720510172503062360713760607985203309940766118866743491802189150471978534404249 * 2^{2299} + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + *d*, *d* = 0, 4, 6, 10, 12, 16 (772 digits, 1/28/2018, Riecoin #822096)

29749903422302373222996698880833194129159047179535887991184960156219652236318921 * 2^{2293} + 679631792885016654160023247517239700227428004849763556497260661860592843345 + *d*, *d* = 0, 4, 6, 10, 12, 16 (770 digits, 12/9/2017, Riecoin #793872)

29696802688480280387313212926526693549449146292085717645262228449092881114972806 * 2^{2290} + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + *d*, *d* = 0, 4, 6, 10, 12, 16 (769 digits, 2/25/2018, Riecoin #838224)

29744205023784420961031622414734790416939049568996819659808238403983863222665068 * 2^{2288} + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + *d*, *d* = 0, 4, 6, 10, 12, 16 (769 digits, 2/18/2018, Riecoin #834192)

29707412718946949415029080194980493978605678414396606766712262274235284928962561 * 2^{2278} + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + *d*, *d* = 0, 4, 6, 10, 12, 16 (766 digits, 1/14/2018, Riecoin #814032)

29696978890366869883141509418765838581871522982358338407613039711378021084519043 * 2^{2259} + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + *d*, *d* = 0, 4, 6, 10, 12, 16 (760 digits, 12/31/2017, Riecoin #805968)

29691575669072177222494655186416928710256802541243921484227880404600991044790342 * 2^{2259} + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + *d*, *d* = 0, 4, 6, 10, 12, 16 (760 digits, 12/16/2017, Riecoin #797904)

29738370152765841200477916368997470863233149039979929714395166089470825913521999 * 2^{2250} + 3267273123746637724423731592929240166353975680818870504129389950929427468581 + *d*, *d* = 0, 4, 6, 10, 12, 16 (757 digits, 2/11/2018, Riecoin #830160)

**32821868878860201045633341031688415601401701228
32878265333984717524446848642006351778066196724473
92249620201536539259942023218972369026762290403609
01005487309186655777663859063397693729163631275766
07799875309038457637116938538279395260265064447747
74261236889041020217108597484837589978261046949778
71991825164994665583879769659044973939714534960362
41885200541893611077817261813672809971503287259089 * 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20** (527 digits, 16 Jun 2019, Vidar Nakling, RIEMINER0.9, PRIMO)

115828580393941*1200# + 5132201 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (515 digits, 18 Jan 2018, Norman Luhn, PRIMO)

4733578067069 * 940# + 626609 + *d*, *d* = 0, 2, 8, 12, 14, 18, 20 (402 digits, May 2016, Norman Luhn)

10^319 + 2219844666811981651 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (320 digits, 9 Apr 2019, Norman Luhn, PRIMO)

4079068497377 * 739# / 14 + *d*, *d* = −4, −2, 2, 4, 8, 14, 16 (319 digits, Dec 2013, Serge Batalov, OpenPFGW, NEWPGEN, PRIMO)

771620215080738 * 700# + 23983691 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (305 digits, 4 Jul 2017, Norman Luhn, VFYPR)

375681809246516 * 700# + 23983691 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (305 digits, 4 Jul 2017, Norman Luhn, VFYPR)

359378518392551 * 700# + 23983691 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)

334063689033226 * 700# + 23983691 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)

295031928451848 * 700# + 23983691 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20 (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)

**359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26** (304 digits, 4 Jul 2017, Norman Luhn, VFYPR)

29995576270632 * 550# + 1277 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26 (236 digits, Jun 2014, Norman Luhn)

330846961 * 503# + 349129635971 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26 (218 digits, Feb 2008, Jens Kruse Andersen)

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

981580217350274 * 460# + 1277 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)

860919759693785 * 460# + 1277 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)

768132737748133 * 460# + 1277 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)

663579549486449 * 460# + 1277 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)

550907700932667 * 460# + 1277 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)

527840551204869 * 460# + 1277 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26 (203 digits, 16 Mar 2017, Norman Luhn)

**663579549486449 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30** (203 digits, 16 Mar 2017, Norman Luhn)

68663510211259 * 337# + 88789 + *d*, *d* = 0, 4, 10, 12, 18, 22, 24, 28, 30 (150 digits, Jan 2010, Norman Luhn)

3336884 * 331# + 80877403191701 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30 (140 digits, Sep 2007, Dirk Augustin & Jens Kruse Andersen)

851437873414817 * 300# + 29247917 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26, 30 (136 digits, 9 Feb 2017, Norman Luhn)

772556746441918 * 300# + 29247919 + *d*, *d* = 0, 4, 10, 12, 18, 22, 24, 28, 30 (136 digits, 9 Feb 2017, Norman Luhn)

772556746441918 * 300# + 29247917 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26, 30 (136 digits, 9 Feb 2017, Norman Luhn)

394833958615791 * 300# + 29247917 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26, 30 (135 digits, 9 Feb 2017, Norman Luhn)

106345403186416 * 300# + 29247913 + *d*, *d* = 0, 4, 6, 10, 16, 18, 24, 28, 30 (135 digits, 9 Feb 2017, Norman Luhn)

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

7644 * 281# + 355388175685385651 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30 (120 digits, May 2016, Roger Thompson)

**772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32** (136 digits, 9 Feb 2017, Norman Luhn)

7425 * 281# + 471487291717627721 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (120 digits, May 2016, Roger Thompson)

118557188915212 * 260# + 25658441 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (118 digits, Jun 2014, Norman Luhn)

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 9268026349694711 * 229# + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (108 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 * 229# + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (107 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4265997347677925 * 229# + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (107 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

2^{345} − (2^{345} mod 193#) + 21623648394477 * 193# + 27196901 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, 12 Sep 2018, Craig Loizides)

2^{345} − (2^{345} mod 191#) + 720445513799051 * 191# + 73018061 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, 12 Sep 2018, Craig Loizides)

2^{345} − (2^{345} mod 191#) + 529379277568620 * 191# + 68187851 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, 12 Sep 2018, Craig Loizides)

2^{345} − (2^{345} mod 191#) + 446242671125040 * 191# + 68187851 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (104 digits, 12 Sep 2018, Craig Loizides)

**13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36** (107 digits, 28 May 2019, Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO)

2^{345} − (2^{345} mod 191#) + 720445513799051 * 191# + 73018061 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (104 digits, 12 Sep 2018, Craig Loizides)

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

613176722801194*151# + 177321217 + *d*, *d* = 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)

446098440707057*151# + 177321217 + *d*, *d* = 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (75 digits, Sep 2014, Michael Stocker, PRIMO)

422969185886875 * 151# + 177321223 + *d*, *d* = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (74 digits, Aug 2014, Michael Stocker)

705484555578416 * 150# + 23378471 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (73 digits, Oct 2014, Norman Luhn)

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

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

467756 * 151# + 193828829641176461 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (66 digits, May 2014, Roger Thompson)

**613176722801194*151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42** (75 digits, Sep 2014, Michael Stocker, PRIMO)

467756 * 151# + 193828829641176461 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (66 digits, May 2014, Roger Thompson)

59125383480754 * 113# + 12455557957 + *d*, *d* = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (61 digits, Sep 2013, Michael Stocker)

78989413043158 * 109# + 38458151 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (59 digits, Jan 2010, Norman Luhn)

450725899 * 113# + 1748520218561 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (56 digits, Nov 2014, Martin Raab)

14815550 * 107# + 4385574275277311 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (50 digits, Feb 2013, Roger Thompson)

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

839858 * 107# + 2566964683459061 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (49 digits, Aug 2009, Jens Kruse Andersen)

337712 * 107# + 3440354553191441 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (48 digits, Aug 2009, Jens Kruse Andersen)

122428 * 107# + 4540852852571921 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (48 digits, Aug 2009, Jens Kruse Andersen)

**4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48** (61 digits, 23 Mar 2017, Norman Luhn)

14815550 * 107# + 4385574275277311 + *d*, *d* = 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (50 digits, Feb 2013, Roger Thompson)

14815550 * 107# + 4385574275277311 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (50 digits, Feb 2013, Roger Thompson)

61571 * 107# + 4803194122972361 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (48 digits, Aug 2009, Jens Kruse Andersen)

381955327397348*80# + 18393209 + *d*, *d* = 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)

381955327397348*80# + 18393209 + *d*, *d* = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (46 digits, Dec 2007, Norman Luhn)

1955206838 * 73# + 44208109063 + *d*, *d* = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (38 digits, Aug 2012, Martin Raab)

322255 * 73# + 1354238543317302647 + *d*, *d* = 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (35 digits, 18 Nov 2016, Roger Thompson)

1464893944 * 67# + 42166182984041 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 (35 digits, Jul 2012, Martin Raab)

457308940 * 67# + 4122369405991 + *d*, *d* = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 (34 digits, Mar 2011, Martin Raab)

**14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50** (50 digits, Feb 2013, Roger Thompson)

381955327397348*80# + 18393209 + *d*, *d* = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (46 digits, Dec 2007, Norman Luhn)

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

108804167016152508211944400342691 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

107173714602413868775303366934621 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

101885197790002105359911556070661 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

101803109763079694387921584406441 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

101047123513223569167212934432341 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

100859765410802682029505696121301 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

100496797396678760339871075201851 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (33 digits, Apr 2008, Jens Kruse Andersen)

**33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56** (40 digits, 25 Jan 2017, Norman Luhn)

322255 * 73# + 1354238543317302647 + *d*, *d* = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (35 digits, 18 Nov 2016, Roger Thompson)

10004646546202610858599716515809907 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (35 digits, Sep 2012, Roger Thompson)

107173714602413868775303366934621 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (33 digits, Apr 2008, Jens Kruse Andersen)

99999999948164978600250563546400 + *d*, *d* = 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67 (32 digits, Nov 2004, Joerg Waldvogel and Peter Leikauf)

1251030012595955901312188450381 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)

1100916249233879857334075234831 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (31 digits, Oct 2003, Hans Rosenthal & Jens Kruse Andersen)

1003234871202624616703163933853 + *d*, *d* = 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)

999999999962618227626700812281 + *d*, *d* = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 (30 digits, Nov 2000, Joerg Waldvogel & Peter Leikauf)

100845391935878564991556707107 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (30 digits, Feb 2013, Roger Thompson)

**322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60** (35 digits, 18 Nov 2016, Roger Thompson)

1003234871202624616703163933853 + *d*, *d* = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (31 digits, Aug 2012, Roger Thompson)

11413975438568556104209245223 + *d*, *d* = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 (29 digits, Jan 2012, Roger Thompson)

5867208169546174917450987997 + *d*, *d* = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

5621078036155517013724659007 + *d*, *d* = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4668263977931056970475231217 + *d*, *d* = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4652363394518920290108071167 + *d*, *d* = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4483200447126419500533043987 + *d*, *d* = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

3361885098594416802447362317 + *d*, *d* = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)

3261917553005305074228431077 + *d*, *d* = 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)

**100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66** (30 digits, Feb 2013, Roger Thompson)

11413975438568556104209245223 + *d*, *d* = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 (29 digits, Jan 2012, Roger Thompson)

11410793439953412180643704677 + *d*, *d* = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 (29 digits, Jan 2012, Roger Thompson)

5867208169546174917450987997 + *d*, *d* = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

5867208169546174917450987997 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

5621078036155517013724659007 + *d*, *d* = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

5621078036155517013724659007 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4668263977931056970475231217 + *d*, *d* = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4668263977931056970475231217 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4652363394518920290108071167 + *d*, *d* = 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

**5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70** (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

5621078036155517013724659007 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Mar 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4668263977931056970475231217 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4652363394518920290108071167 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

4483200447126419500533043987 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jan 2014, Raanan Chermoni & Jaroslaw Wroblewski)

3361885098594416802447362317 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)

3261917553005305074228431077 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)

3176488693054534709318830357 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Jul 2013, Raanan Chermoni & Jaroslaw Wroblewski)

2650778861583720495199114537 + *d*, *d* = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 (28 digits, Feb 2013, Raanan Chermoni & Jaroslaw Wroblewski)

2406179998282157386567481191 + *d*, *d* = 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)

**622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76** (30 digits, December 27, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

248283957683772055928836513589 + *d*, *d* = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)

138433730977092118055599751669 + *d*, *d* = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski)

39433867730216371575457664399 + *d*, *d* = 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (29 digits, 8 Jan 2015, Raanan Chermoni & Jaroslaw Wroblewski)

2406179998282157386567481191 + *d*, *d* = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)

2348190884512663974906615481 + *d*, *d* = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (28 digits, Dec 2012, Raanan Chermoni & Jaroslaw Wroblewski)

917810189564189435979968491 + *d*, *d* = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, May 2011, Raanan Chermoni & Jaroslaw Wroblewski)

656632460108426841186109951 + *d*, *d* = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, 19 Feb 2011, Raanan Chermoni & Jaroslaw Wroblewski)

630134041802574490482213901 + *d*, *d* = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 (27 digits, 9 Feb 2011, Raanan Chermoni & Jaroslaw Wroblewski)

{37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}

**701870455949526598513130862539 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80** (30 digits, April 7, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

686962597479437604159786541481 + *d*, *d* = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, April 27, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

667424014858149638371951648871 + *d*, *d* = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, February 18, 2019, Raanan Chermoni & Jaroslaw Wroblewski)

639121700726230052098229452019 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, December 23, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

622803914376064301858782434517 + *d*, *d* = 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84 (30 digits, December 27, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

594750459626903777773717631519 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, December 23, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

593820854957327357933627374349 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, December 23, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

562422394447827908154562532159 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, August 30, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

562324418721793120042174985351 + *d*, *d* = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 (30 digits, August 30, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

494179332730633784520908832239 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 (30 digits, 23 May 2018, Raanan Chermoni & Jaroslaw Wroblewski)

**622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84** (30 digits, December 27, 2018, Raanan Chermoni & Jaroslaw Wroblewski)

248283957683772055928836513589 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 1 Aug 2016, Raanan Chermoni & Jaroslaw Wroblewski)

138433730977092118055599751669 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (30 digits, 8 Oct 2015, Raanan Chermoni & Jaroslaw Wroblewski)

39433867730216371575457664399 + *d*, *d* = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 (29 digits, 8 Jan 2015, Raanan Chermoni & Jaroslaw Wroblewski)

{29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}

The largest known prime k-tuplets | ||||

k |
Digits | Prime k-tuplet |
Who | When |

1 | 24862048 | 2^{82589933} − 1 |
P. Laroche, G. Woltman, S. Kurowski, A. Blosser, et al (GIMPS) |
21 Dec 2018 |

2 | 388342 | 2996863034895 * 2^{1290000} ± 1 |
Tom Greer, TWINGEN, PRIMEGRID, LLR | Sep 2016 |

3 | 20008 | 4111286921397 * 2^{66420} + d, d = −1, 1, 5 |
Peter Kaiser, POLYSIEVE, LLR, PRIMO | 24 Apr 2019 |

4 | 10132 | 667674063382677 * 2^{33608} + d, d = −1, 1, 5, 7 |
Peter Kaiser, PRIMO | 27 Feb 2019 |

5 | 1606 | 394254311495 * 3733# / 2 + d, d = -8, -4, -2, 2, 4 |
Serge Batalov, NEWPGEN, OPENPFGW, PRIMO | Nov 2017 |

6 | 1037 | 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 |
Norman Luhn, APSIEVE, PRIMO | 14 Mar 2016 |

7 | 527 | 32821868878860201045633341031688415601401701228
32878265333984717524446848642006351778066196724473
92249620201536539259942023218972369026762290403609
01005487309186655777663859063397693729163631275766
07799875309038457637116938538279395260265064447747
74261236889041020217108597484837589978261046949778
71991825164994665583879769659044973939714534960362
41885200541893611077817261813672809971503287259089 * 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 |
Vidar Nakling, RIEMINER0.9, PRIMO | 16 Jun 2019 |

8 | 304 | 359378518392551 * 700# + 23983691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 |
Norman Luhn, VFYPR | 4 Jul 2017 |

9 | 203 | 663579549486449 * 460# + 1277 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30 |
Norman Luhn | Mar 2017 |

10 | 136 | 772556746441918 * 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 |
Norman Luhn | 9 Feb 2017 |

11 | 107 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 * 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
Peter Kaiser, David Stevens, POLYSIEVE, PFGW, PRIMO | 28 May 2019 |

12 | 75 | 613176722801194*151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 |
Michael Stocker, PRIMO | Sep 2014 |

13 | 61 | 4135997219394611 * 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 |
Norman Luhn | 23 Mar 2017 |

14 | 50 | 14815550 * 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 |
Roger Thompson | Feb 2013 |

15 | 40 | 33554294028531569*61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 |
Norman Luhn | 25 Jan 2017 |

16 | 35 | 322255 * 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 |
Roger Thompson | 18 Nov 2016 |

17 | 30 | 100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 |
Roger Thompson | Feb 2013 |

18 | 28 | 5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 |
Raanan Chermoni & Jaroslaw Wroblewski | Mar 2014 |

19 | 30 | 622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76 |
Raanan Chermoni & Jaroslaw Wroblewski | 27 Dec 2018 |

20 | 30 | 701870455949526598513130862539 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 |
Raanan Chermoni & Jaroslaw Wroblewski | 7 Apr 2019 |

21 | 30 | 622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76, 82, 84 |
Raanan Chermoni & Jaroslaw Wroblewski | 27 Dec 2018 |

22 | 2 | {23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |

23 | 2 | {19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113} | - | - |

24 | - | There are no known prime 24-tuplets | - | - |

First appearance of a non-trivial prime k-tuplet | ||||

k |
Digits | Prime k-tuplet |
Who | When |

<12 | - | No reliable information | - | - |

12 | 13 | 1418575498567 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 |
D. Betsis & S. Säfholm | 1982 |

13 | 14 | 10527733922579 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 |
D. Betsis & S. Säfholm | 1982 |

14 | 17 | 21817283854511261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 |
D. Betsis & S. Säfholm | 1982 |

15 | 21 | 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 |
Joerg Waldvogel | 1996 |

16 | 21 | 347709450746519734877 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 |
Joerg Waldvogel | 1996 |

17 | 22 | 1620784518619319025971 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 |
Joerg Waldvogel | 1997 |

18 | 25 | 2845372542509911868266807 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 |
Joerg Waldvogel & Peter Leikauf | 14 Nov 2000 |

19 | 27 | 630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 |
Raanan Chermoni & Jaroslaw Wroblewski | 9 Feb 2011 |

20 | 28 | 3941119827895253385301920029 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 |
Raanan Chermoni & Jaroslaw Wroblewski | 24 Jun 2014 |

21 | 29 | 39433867730216371575457664399 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 |
Raanan Chermoni & Jaroslaw Wroblewski | 8 Jan 2015 |

First appearance of 100 digits | ||||

k |
Digits | Prime k-tuplet |
Who | When |

1 | 157 | 2^{521} − 1 |
R. M. Robinson | Jan 1952 |

2-5 | - | No reliable information | - | - |

6 | 133 | 2 * 10^{132} + 75543532187 + d, d = 0, 4, 6, 10, 12, 16 |
Tony Forbes | Apr 1994 |

7 | 104 | 4293326603 * 233# + 399389 + d, d = 0, 2, 8, 12, 14, 18, 20 |
Radoslaw Naleczynski | Dec 1998 |

8 | 110 | 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 |
Norman Luhn | Feb 2001 |

9 | 110 | 388793398651 * 250# + 1042090781 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 |
Norman Luhn | Feb 2001 |

10 | 103 | 72613488698235 * 227# + 39058751 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 |
Norman Luhn | Apr 2004 |

11 | 104 | 24698258 * 239# + 28606476153371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
Norman Luhn & Jens Kruse Andersen | Aug 2004 |

First appearance of 1000 digits | ||||

k |
Digits | Prime k-tuplet |
Who | When |

1 | 1332 | 2^{4423} − 1 |
Alexander Hurwitz | Nov 1961 |

2 | 1040 | 256200945 * 2^{3426} ± 1 |
Oliver Atkin & N. W. Rickert | 1980 |

3 | 1083 | 437850590*(2^{3567} − 2^{1189}) − 6*2^{1189} +
d, d = −5, −1, 1 |
Tony Forbes | Dec 1996 |

4 | 1004 | 76912895956636885*(2^{3279} − 2^{1093}) − 6*2^{1093} +
d, d = −7, −5, −1, 1 |
Tony Forbes | Sep 1998 |

5 | 1034 | 31969211688*2400# + 16061 + d, d = 0, 2, 6, 8, 12 |
Norman Luhn | Jul 2002 |

6 | 1037 | 28993093368077 * 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 |
Norman Luhn, APSIEVE, PRIMO | 14 Mar 2016 |

First appearance of 10000 digits | ||||

k |
Digits | Prime k-tuplet |
Who | When |

1 | 13395 | 2^{44497} − 1 |
Harry Nelson & David Slowinski | Apr 1979 |

2 | 11713 | 242206083 * 2^{38880} ± 1 |
H. K. Indlekofer & A. Járai | Nov 1995 |

3 | 10047 | 2072644824759 * 2^{33333} + d, d = −1, 1, 5 |
Norman Luhn, François Morain, FastECPP | Nov 2008 |

4 | 10132 | 667674063382677 * 2^{33608} + d, d = −1, 1, 5, 7 |
Peter Kaiser, PRIMO | 27 Feb 2019 |

First appearance of 100000 digits | ||||

k |
Digits | Prime k-tuplet |
Who | When |

1 | 227832 | 2^{756839} − 1 |
David Slowinski & Paul Gage | Apr 1992 |

2 | 100355 | 65516468355 * 2^{333333} ± 1 |
Peter Kaiser, NEWGEN, PRIMEGRID, TPS, LLR | Aug 2009 |

First appearance of 1000000 digits | ||||

k |
Digits | Prime k-tuplet |
Who | When |

1 | 2098960 | 2^{6972593} − 1 |
Nayan Hajratwala, George Woltman, Scott Kurowski et al (GIMPS) |
Jun 1999 |

First appearance of 10000000 digits | ||||

k |
Digits | Prime k-tuplet |
Who | When |

1 | 12978189 | 2^{43112609} − 1 |
Edson Smith, George Woltman, Scott Kurowski et al (GIMPS) |
Sep 2008 |

List of all possible patterns of prime *k*-tuplets

List of the *smallest* prime *k*-tuplets

Various lists of prime k-tuplets

Near misses: Clusters of primes that didn't quite make it into the main list

The Hardy-Littlewood constants pertaining to the distribution of prime *k*-tuplets [HL22]

Jens Kruse Andersen: The Largest Known Simultaneous Primes

Jens Kruse Andersen: Consecutive Primes in Arithmetic Progression

Jens Kruse Andersen: Largest Consecutive Factorizations

Dirk Augustin: Cunningham Chain records

Chris K. Caldwell: The Largest Known Primes

Chris K. Caldwell: Top twenty twin primes

TF: Ten consecutive primes in arithmetic progression

Dr. Minh. L. Perez Press: Smarandache Primes

N. J. A. Sloane: On-Line Encyclopedia of Integer Sequences

Manfred Toplic: The Nine and Ten Primes Project

Robin Whitty: Theorem of the Day

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

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, proven primes. Numbers which have merely passed
the Fermat test, *a*^{N} = *a* (mod *N*), will need to
be validated. If *N* − 1 or *N* + 1 is sufficiently factorized (usually
just under a third), the methods of Brillhart, Lehmer and Selfridge [BLS75] will
suffice. Otherwise the numbers may have to be subjected to a general primality
test, such as the Jacobi sum test of Adleman, Pomerance, Rumely, Cohen and
Lenstra (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/*p*^{2}) =
∑_{n = 1 to ∞} 1/*n*^{2} = π^{2}/6.

But π^{2} is irrational; so the product on the left cannot have a finite
number of factors.

In its simplest form, the *prime number theorem* states that the
number of primes less than *x* is asymptotic to *x*/(log *x*).
This was first proved by Hadamard and independently by de la Vallee Poussin in
1896. Later, de la Vallee Poussin found a better estimate:

∫_{u = 0 to x} d*u*/(log *u*) + error term,

where the error term is bounded above by
*A* *x* exp(−*B* √(log *x*)) for some constants
*A* and *B*. With more work (H.-E. Richert, 1967), √(log *x*)
in this last expression can be replaced by
(log *x*)^{3/5}(log log *x*)^{−1/5}.
The most important unsolved conjecture of prime number theory, indeed, all of mathematics
- the Riemann Hypothesis - asserts that the error term can be bounded by a function
of the form *A* √*x* log *x*.

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

2 *C*_{2} *x* / (log *x*)^{2},

where
*C*_{2} = ∏_{p prime, p > 2} * p*(*p* − 2) / (*p* − 1)^{2}
= 0.66016... is known as the *twin prime constant*.

V. Brun showed that the sequence of twins is thin enough for the series
∑_{p and p + 2 prime} 1 / *p* to converge. The *twin prime conjecture*
states that the sum has infinitely many terms. The nearest to proving the
conjecture is 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* + *b _{k}*

*H _{k}*

*where*

*H _{k}* = ∏

*C _{k}* = ∏

*v* = *v*(*p*)* is the number of distinct remainders of*
*b*_{1}, *b*_{2}, ..., *b** _{k} modulo p and D is the product of the
differences |b_{i} − b_{j}|*, 1 ≤

The first product in *H _{k}* is over the primes not greater than

It is worth pointing out that with modern mathematical software
the prime *k*-tuplet constants *C _{k}* can be determined to great accuracy.
The way not to do it is to use the defining formula. Unless you are very patient,
calculating the product over a sufficient number of primes for, say, 20 decimal place accuracy
would not be feasible. Instead there is a useful transformation originating from the product
formula for the Riemann ζ function:

log *C _{k}* =
− ∑

[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.

[F97f] Tony Forbes, Prime 17-tuplet, NMBRTHRY Mailing List, September 1997.

[F02] Tony Forbes, Titanic prime quintuplets,
*M500* **189** (December, 2002), 12-13.

[F09] Tony Forbes, Gigantic prime triplets,
*M500* **226** (February, 2009), 18-19.

[Guy94] Richard K. Guy, *Unsolved Problems in Number Theory*, second edn.,
Springer-Verlag, New York 1994.

[HL22] G. H. Hardy and J. E. Littlewood, Some problems of *Partitio Numerorum*: III;
on the expression of a number as a sum of primes, *Acta Mathematica* **44** (1922), 1-70.

[HR73] H. Halberstam and H.-E Richert, *Sieve Methods*, Academic Press, London 1973.

[Rib95] P. Ribenboim, *The New Book of Prime Number Records*, 3rd edn.,
Springer-Verlag, New York 1995

[R95] Warut Roonguthai, Prime quadruplets, NMBRTHRY Mailing List, September 1995.

[R96a] Warut Roonguthai, Prime quadruplets, *M500* **148** (February 1996), 9.

[R96b] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1996.

[R96c] Warut Roonguthai, Large prime quadruplets, *M500*, **153** (December, 1996), 4-5.

[R97a] Warut Roonguthai, Large prime quadruplets, NMBRTHRY Mailing List, September 1997.

[R97b] Warut Roonguthai, Large prime quadruplets, *M500* **158** (November 1997), 15.

[S95] G. John Stevens, Prime Quadruplets, *J. Recr. Math.* **27** (1995), 17-22.