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.
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 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, 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, 5, 6, 7, 8, 9,10, 11, 12, 13, 14, 15 and 16. I do not know of any prime k-tuplets for k greater than 16, except the trivial ones that occur near the beginning of the prime number sequence. Nor do I know of any prime 14-, 15- and 16-tuplets apart from the ones listed in this page.
The symbol for multiplication is an asterisk: x*y means x times y. The symbol "^" means "to the power of": Thus x^2 = x*x, x^3 = x*x*x and so on.
For k > 2, the abbreviation N + b1, b2, ..., bk denotes the k numbers {N + b1, N + b2, ..., N + bk}. Prime twins are represented by N +/- 1, which is short for N plus one and N minus one.
I also use the notation p# of Caldwell and Dubner [CD93] as a convenient shorthand for 2*3*5*...*p, the product of all the primes up to and including p.
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, Tony Forbes. You can see what I mean by 'large' by studying the lists. If the numbers are not too big, say up to 200 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.
242206083 * 2^38880 +/- 1 (11713 digits, 1995, K-H. Indlekofer & A. Ja'rai)
570918348 * 10^5120 +/- 1 (5129 digits, 1995, Harvey Dubner)
697053813 * 2^16352 +/- 1 (4932 digits, 1994, K-H. Indlekofer & A. Ja'rai)
6797727 * 2^15328 +/- 1 (4622 digits, 1995, Tony Forbes [F95b, F97c])
1691232 * 1001 * 10^4020 +/- 1 (4030 digits, 1993, H. Dubner)
4650828 * 1001 * 10^3429 +/- 1 (3429 digits, 1993, H. Dubner)
1706595 * 2^11235 +/- 1 (3389 digits, 1990, B. Parady, J. Smith, S. Zarantonello)
459 * 2^8529 +/- 1 (2571 digits, 1993, H. Dubner)
1171452282 * 10^2490 +/- 1 (2500 digits, H. Dubner)
571305 * 2^7701 +/- 1 (2324 digits, 1990, B. Parady, J. Smith, S. Zarantonello)
See Chris Caldwell, The Largest Known Primes, for further details and full references.
437850590*(2^3567 - 2^1189) - 6*2^1189 - 5, -1, +1 (1083 digits, 1996, Tony Forbes [F96e, F97a])
2^2400 + 14906370057 + 0, 4, 6 (723 digits, 1994, T. Forbes, [F94b, F96a])
2^2080 + 17304485517 + 0, 4, 6 (627 digits, 1994, T. Forbes, [F94b, F96a])
2^2080 + 9595427487 + 0, 4, 6 (627 digits, 1994, T. Forbes, [F94b, F96a])
2^2016 + 2294808807 + 0, 4, 6 (607 digits, 1994, T. Forbes [F94b])
2^1664 + 3405302217 + 0, 4, 6 (501 digits, 1994, T. Forbes [F94b])
10^499 + 883750143961 + 2, 6, 8 (500 digits, 1996, Warut Roonguthai [R96b, R96c])
10^499 + 883750143961 + 0, 2, 6 (500 digits, 1996, Warut Roonguthai [R96b, R96c])
4 * 10^400 + 6368335473 + 0, 4, 6 (401 digits, 1994, T. Forbes [F94b])
10^399 + 34993836001 + 2, 6, 8 (400 digits, 1995, Warut Roonguthai [R95, R96a])
10^499 + 883750143961 + 0, 2, 6, 8 (500 digits, 1996, Warut Roonguthai [R96b, R96c])
10^399 + 34993836001 + 0, 2, 6, 8 (400 digits, 1995, Warut Roonguthai [R95, R96a])
6*2^676*(8389700*R + (2^-676 mod R)) + 1, -1, -5, -7, R = 11*13*17*...*271 (320 digits, 1995, A. O. L. Atkin)
2^1056 + 1301655396715 + 0, 2, 6, 8 (318 digits, 1994, T. Forbes [F94c, F96a])
10^299 + 175915846201 + 0, 2, 6, 8 (300 digits, 1995, Warut Roonguthai)
10^299 + 140159459341 + 0, 2, 6, 8 (300 digits, 1995, Warut Roonguthai [R95, R96a])
10^250 + 4451442*541# + 271# + 199# + 103# + 9699751 + 0, 2, 6, 8 (1997, Chad Davis)
2^800 + 10169432335 + 0, 2, 6, 8 (241 digits, 1994, T. Forbes [F94c, F96a])
8947613442 * 53# * 2^672 + 101 + 0, 2, 6, 8 [, 12] (232 digits, 1997, A.O.L. Atkin)
2^768 + 97745478256435 + 0, 2, 6, 8 (232 digits, 1997, T. Forbes)
8947613442 * 53# * 2^672 + 101 + 0, 2, 6, 8, 12 (232 digits, 1997, A.O.L. Atkin)
2^768 + 108676793455855 + 0, 2, 6, 8, 12 (232 digits, 1997, T. Forbes)
87588293129 * 43# * 2^511 + 16057 + 0, 4, 6, 10, 12 (181 digits, 1997, A.O.L. Atkin)
36181550775 * 43# * 2^512 + 16057 + 4, 6, 10, 12, 16 (181 digits, 1997, A.O.L. Atkin)
16857474975 * 43# * 2^512 + 97 + 0, 4, 6, 10, 12 (181 digits, 1997, A.O.L. Atkin)
31579271047 * 43# * 2^ 511 + 16057 + 0, 4, 6, 10, 12 (181 digits, 1997, A.O.L. Atkin)
11970383561 * 43# * 2^512 + 97 + 0, 4, 6, 10, 12 (181 digits, 1997, A.O.L. Atkin)
1702757 * 89# * 2^448 + 19421 + 0, 2, 6, 8, 12 (1996, 176 digits, A.O.L. Atkin)
42339482 * 61# * 2^480 + 101 + 0, 2, 6, 8, 12 (1996, 176 digits, A.O.L. Atkin)
2^576 + 79801763715655 + 0, 2, 6, 8, 12 (174 digits, 1995, T. Forbes, [F95c, F96a])
82248305245 * 43# * 2^479 + 16057 + 0, 4, 6, 10, 12, 16 (172 digits, 1997, A.O.L. Atkin)
2^512 + 6638977280721 + 0, 4, 6, 10, 12, 16 (155 digits, 1996, T. Forbes [F96d, F96f])
2 * 10^132 + 75543532187 + 0, 4, 6, 10, 12, 16 (1994, T. Forbes [F94a, F96a])
29964012*151# + 89# + 97 + 0, 4, 6, 10, 12, 16 (67 digits, 1997, Chad Davis)
10^45 + 22172544387 + 0, 4, 6, 10, 12, 16 (1997, Chad Davis)
5 * 10^42 + 1210414367 + 0, 4, 6, 10, 12, 16 (1994, T. Forbes [F93])
10^30 + 5773122297 + 0, 4, 6, 10, 12, 16 (1997, Chad Davis)
10^29 + 77882127 + 0, 4, 6, 10, 12, 16 (1993, T. Forbes [F94a])
10^20 + 4560084057 + 0, 4, 6, 10, 12, 16 (1997, Chad Davis)
10^20 + 4445040387 + 0, 4, 6, 10, 12, 16 (1997, Chad Davis)
426955143694213197848463574726328636553002998029907759380111141003679237691 + 0, 2, 6, 8, 12, 18, 20 (75 digits, 1997, A.O.L. Atkin)
582994762600347672560616756460401857268344059297139419451 + 0, 2, 6, 8, 12, 18, 20 (57 digits, 1996, A.O.L. Atkin)
15 * 10^52 + 8496272339051 + 0, 2, 6, 8, 12, 18, 20 (1995, T. Forbes, [F95a])
11456782178002488855779277536193082378054961 + 0, 2, 6, 8, 12, 18, 20 (44 digits, 1996, A.O.L. Atkin)
22 * 10^38 + 2251884085366561 + 0, 2, 6, 8, 12, 18, 20 (1996, T. Forbes)
22 * 10^38 + 2241278889512329 + 0, 2, 8, 12, 14, 18, 20 (1996, T. Forbes)
22 * 10^38 + 2166839792687509 + 0, 2, 8, 12, 14, 18, 20 (1996, T. Forbes)
22 * 10^38 + 2027415237697789 + 0, 2, 8, 12, 14, 18, 20 (1996, T. Forbes)
22 * 10^38 + 1997262086363329 + 0, 2, 8, 12, 14, 18, 20 (1996, T. Forbes)
22 * 10^38 + 1836046440291751 + 0, 2, 6, 8, 12, 18, 20 (1996, T. Forbes)
582994762600347672560616756460401857268344059297139419451 + 0, 2, 6, 8, 12, 18, 20, 26 (57 digits, 1996, A.O.L. Atkin)
15 * 10^52 + 11527947572567 + 0, 2, 6, 12, 14, 20, 24, 26 (1995, T. Forbes, [F95a, F96a])
9 * 10^50 + 3219584035187 + 0, 2, 6, 12, 14, 20, 24, 26 (1994, T. Forbes, [F94a, F96a])
11456782178002488855779277536193082378054961 + 6, 8, 12, 18, 20, 26, 30, 32 (44 digits, 1996, A.O.L. Atkin)
11456782178002488855779277536193082378054961 + 0, 2, 6, 8, 12, 18, 20, 26 (44 digits, 1996, A.O.L. Atkin)
22 * 10^38 + 2241278889512323 + 0, 6, 8, 14, 18, 20, 24, 26 (1996, T. Forbes)
22 * 10^38 + 1997262086363323 + 0, 6, 8, 14, 18, 20, 24, 26 (1996, T. Forbes)
22 * 10^38 + 1124308227181297 + 0, 2, 6, 12, 14, 20, 24, 26 (1996, T. Forbes)
22 * 10^38 + 687688537960111 + 0, 2, 6, 8, 12, 18, 20, 26 (1996, T. Forbes)
22 * 10^38 + 632949030380167 + 0, 2, 6, 12, 14, 20, 24, 26 (1996, T. Forbes)
11456782178002488855779277536193082378054961 + 2, 6, 8, 12, 18, 20, 26, 30, 32 (44 digits, 1996, A.O.L. Atkin)
11456782178002488855779277536193082378054961 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (44 digits, 1996, A.O.L. Atkin)
22 * 10^38 + 2241278889512319 + 0, 4, 10, 12, 18, 22, 24, 28, 30 (1996, T. Forbes)
22 * 10^38 + 1997262086363319 + 0, 4, 10, 12, 18, 22, 24, 28, 30 (1996, T. Forbes)
22 * 10^38 + 632949030380167 + 0, 2, 6, 12, 14, 20, 24, 26, 30 (1996, T. Forbes)
22 * 10^35 + 111245519078161 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (1995, T. Forbes [F95b, F96a])
26 * 10^32 + 30697502738421 + 2, 6, 8, 12, 18, 20, 26, 30, 32 (1995, T. Forbes [F95b, F96a])
26 * 10^32 + 30697502738421 + 0, 2, 6, 8, 12, 18, 20, 26, 30 (1995, T. Forbes [F95b, F96a])
16 * 10^32 + 141561203196933 + 0, 4, 6, 10, 16, 18, 24, 28, 30 (1995, T. Forbes)
13 * 10^30 + 117518621487339 + 0, 4, 10, 12, 18, 22, 24, 28, 30 (1995, T. Forbes)
11456782178002488855779277536193082378054961 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (44 digits, 1996, A.O.L. Atkin)
22 * 10^38 + 2241278889512317 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (1996, T. Forbes [F96f])
26 * 10^32 + 30697502738421 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (1995, T. Forbes [F95b, F96a])
12 * 10^28 + 14884968491321 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (1994, T. Forbes [F94a, F96a])
495064300630708278713578451 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 [, 36, 42] (1997, A.O.L. Atkin)
10^25 + 22901748046151047 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (1996, T. Forbes)
10^25 + 22193215840290337 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (1996, T. Forbes)
10^25 + 18571163403848161 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 (1996, T. Forbes)
25 * 10^23 + 4104936514466137 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (1995, T. Forbes [F95b, F96a])
2^80 + 1051069612640371 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 (1995, T. Forbes [F95b, F96a])
495064300630708278713578451 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 [, 42] (1997, A.O.L. Atkin)
40293597000693392110871 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 [, 42] (1997, A.O.L. Atkin)
300006711401831244037 + 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (1995, T. Forbes)
300004226699607434083 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (1995, T. Forbes [F95b, F96a])
300001698148447235173 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (1995, T. Forbes [F95b, F96a])
300001141243282082683 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (1995, T. Forbes [F95b, F96a])
200048950438497184873 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36 (1995, T. Forbes [F95b])
30021077544678155051 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (1997, TF)
20116664101557058151 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (1997, TF)
20033602181940472241 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 (1997, TF)
495064300630708278713578451 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (1997, A.O.L. Atkin)
40293597000693392110871 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (1997, A.O.L. Atkin)
300006711401831244037 + 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (1995, T. Forbes [F95b, F96a])
30021077544678155051 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (1997, T. Forbes)
20033602181940472241 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (1997, TF)
11519177688150777761 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (1997, TF)
11319107721272355839 + 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (1997, TF)
10923754124049321671 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 (1997, T. Forbes)
10756418345074847279 + 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (1997, T. Forbes)
10665851757039303667 + 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 (1997, TF)
964013473328959309238999 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1997, T. Forbes)
237600689963370454477693 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (1997, T. Forbes)
234627938711125459673189 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1997, T. Forbes)
229719405891782686276333 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (1997, T. Forbes)
228447597916849643124809 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1997, T. Forbes)
226918527762847675808069 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1997, T. Forbes)
217566915393744485481929 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1997, T. Forbes)
216630440009481892196819 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 (1997, T. Forbes)
215480372703622118679523 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (1997, T. Forbes)
214211269787751089540623 + 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 (1997, T. Forbes)
11319107721272355839 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 [, 60] (1997, T. Forbes)
10756418345074847279 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (1997, T. Forbes)
6808488664768715759 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (1996, T. Forbes)
6120794469172998449 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (1996, T. Forbes [F96f])
5009128141636113611 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (1996, T. Forbes [F96c, F96d])
79287805466244209 + 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 (1982, Dimitrios Betsis & Sten Säfholm [Guy94, Section A9])
21817283854511261 + 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 (1982, Dimitrios Betsis & Sten Säfholm [Guy94, Section A9])
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61
84244343639633356306067 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 (1997, T. Forbes)
8985208997951457604337 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (1997, T. Forbes)
2088253704394088213987 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (1997, T. Forbes)
1337707385720650557617 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (1997, TF)
205700275761622834847 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (1997, T. Forbes, [F97b])
107862607835977274207 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (1997, T. Forbes)
47710850533373130107 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 [, 60] (1997, T. Forbes [F97d])
36351118555624575707 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 (1997, T. Forbes)
17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73
11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67
47710850533373130107 + 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 (1997, T. Forbes [F97d])
13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73
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
Huen, YK, Goldbach 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 k consecutive prime numbers such that in some sense the difference between the first and the last is as small as possible. The idea is to generalise the concept of prime twins.
More precisely: We first define s(k) to be the smallest number s for which there exist k integers b1 < b2 < ... < bk, bk - b1 = s and, for every prime q, not all the residues modulo q are represented by b1, b2, ..., bk. A prime k-tuplet is then defined as a sequence of consecutive primes {p1, p2, ..., pk}, with pk - p1 = s(k). Observe that the definition excludes a finite number (for each k) of dense clusters at the beginning of the prime number sequence - for example, {97, 101, 103, 107, 109} satisfies the conditions of the definition of a prime 5-tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.
The simplest case is s(2) = 2, corresponding to prime twins: {p, p+2}. Next, s(3) = 6 and two types of prime triplets: {p, p+2, p+6} and {p, p+4, p+6}, followed by s(4) = 8 with just one pattern: {p, p+2, p+6, p+8} of prime quadruplets. The sequence continues with s(5) = 12, s(6) = 16, s(7) = 20, s(8) = 26, s(9) = 30, s(10) = 32, s(11) = 36, s(12) = 42, s(13) = 48, s(14) = 50, s(15) = 56, s(16) = 60, s(17) = 66 and so on. It is number A008407 in N.J.A. Sloane's On-line Encyclopedia of Integer Sequences.
In keeping with similar published lists, I have decided not to accept anything other than true, verifiable primes. Numbers which have merely passed the Fermat test, a^(N-1) = 1 (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), 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 the book) goes like this: We have
product{p prime, 1/(1 - 1 / p^2)} = sum{n=1 to infinity, 1 / n^2} = pi^2/6.
But pi^2 is irrational, so the product on the left cannot have a finite number of factors.
In its simplest form, the prime number theorem states that the number of primes less than x is asymptotic to x/(log x). This was first proved by Hadamard and independently by de la Vallee Poussin in 1896. Later, de la Vallee Poussin found a better estimate:
integral{u=0 to x, du/(log u)} + error term,
where the error term is bounded above by A x exp(-B sqrt(log x)) for some constants A and B. With more work (H.-E. Richert, 1967) the exponent 1/2 of log x in this last expression can be replaced by 3/5. The most important unsolved conjecture of prime number theory, indeed, all of mathematics, - the Riemann Hypothesis - asserts that the error term can be bounded by the function A sqrt(x) log x.
G.H. Hardy & J.E. Littlewood did the first serious work on the distribution of prime twins. In their paper 'Some problems of Partitio Numerorum: III...' [HL22], they conjectured a formula for the number of twins between 1 and x:
2 C2 x / (log x)^2,
where C2 = product{p > 2, p(p-2) / (p-1)} = 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 b1, b2, ..., bk be k distinct integers. Then the number of groups of primes N + b1, N + b2, ..., N + bk, between 2 and x is approximately
H C integral{from u=0 to x, du / (log u)^k},
where
H = product{p <= k, p^(k-1) (p-v) / (p-1)^k}* product{p>k, p|D, (p-v) / (p-k)},
C = product{p > k, p^(k-1) (p-k) / (p-1)^k},
v = v(p) is the number of distinct remainders of b1, b2, ..., bk modulo p and D is the product of the differences of the b's.
The first product in H is over the primes not greater than k, the second is over the primes greater than k which divide D and the product C is over all primes greater than k. If you put k = 2, b1 = 0 and b2 = 2, then v(2) = 1, v(p) = p - 1 for p > 2, H = 2, and C = C2, the twin prime constant given above.
[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.
[F93] Tony Forbes, Prime k-tuplets again, M500, 134 (August, 1993), 14-15.
[F94a] Tony Forbes, Prime k-tuplets - 5, M500, 137 (April, 1994), 12-14.
[F94b] Tony Forbes, Prime k-tuplets - 6, M500, 139 (July, 1994), 14-15.
[F94c] Tony Forbes, Prime k-tuplets - 7, M500, 140 (October, 1994), 16.
[F95a] Tony Forbes, Prime k-tuplets - 8, M500, 145 (July, 1995), 15.
[F95b] Tony Forbes, Prime k-tuplets - 9, M500, 146 (September, 1995), 6-8.
[F95c] Tony Forbes, Prime k-tuplets - 11, M500, 147 (November, 1995), 6.
[F96a] Tony Forbes, A small collection of prime k-tuplets, NMBRTHRY Mailing List, January, 1996.
[F96b] Tony Forbes, Prime k-tuplets - 12, M500, 148 (February, 1996), 9-10.
[F96c] Tony Forbes, Prime 14-tuplet, NMBRTHRY Mailing List, October, 1996.
[F96d] Tony Forbes, Prime k-tuplets - 13, M500, 153 (December, 1996), 5-7.
[F96e] Tony Forbes, Large prime triplets, NMBRTHRY Mailing List, December, 1996.
[F96f] Tony Forbes, Prime k-tuplets, NMBRTHRY Mailing List, December, 1996.
[F97a] Tony Forbes, Prime triplets, M500, 154 (February, 1997), 13.
[F97b] Tony Forbes, Prime 15-tuplet, NMBRTHRY Mailing List, March, 1997.
[F97c] Tony Forbes, A large pair of twin primes, Math. Comp., 66 (January 1997), 541-545.
[F97d] Tony Forbes, Prime 16-tuplet, NMBRTHRY Mailing List, May, 1997.
[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.
[S95] G. John Stevens, Prime Quadruplets, J. Recr. Math., 27 (1995), 17-22.
Classification: SCIENCE / MATHEMATICS / NUMBERS / PRIME NUMBERS / DISTRIBUTION OF PRIME NUMBERS.
URL: http://www.ltkz.demon.co.uk/ktuplets.htm
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, 1997. Email tonyforbes@ltkz.demon.co.uk
Last updated: 26 June 1997.