Abstract
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 (1997- Aug 2021); anthony.d.forbes@gmail.com.
Additions
Recent additionsContents
Prime Numbers
Prime numbers are the building blocks of arithmetic. They are a special type of number because they cannot be broken down into smaller factors. 13 is prime because 13 is 1 times 13 (or 13 times 1), and that's it. There's no other way of expressing 13 as something times something. On the other hand, 12 is not prime because it splits into 2 times 6, or 3 times 4.
The first prime is 2. The next is 3. Then 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, 467, 479, 487, 491, 499, 503 and so on. If you look at the first 10000 primes, you will see a list of numbers with no obvious pattern. There is even an air of mystery about them; if you didn't know they were prime numbers, you would probably have no idea how to continue the sequence. Indeed, if you do manage discover a simple pattern, you will have succeeded where some of the finest brains of all time have failed. For this is an area where mathematicians are well and truly baffled.
We do know fair amount about prime numbers, and an excellent starting point if you want to learn more about the subject is Chris Caldwell's web site: The Largest Known Primes. We know that the sequence of primes goes on for ever. We know that it thins out. The further you go, the rarer they get. We even have a simple formula for estimating roughly how many primes there are up to some large number without having to count them one by one. However, even though prime numbers have been the object of intense study by mathematicians for hundreds of years, there are still fairly basic questions which remain unanswered.
Prime Twins
If you look down the list of primes, you will quite often see two consecutive odd numbers, like 3 and 5, 5 and 7, 11 and 13, 17 and 19, or 29 and 31. We call these pairs of prime numbers {p, p + 2} prime twins.
The evidence suggests that, however far along the list of primes you care to look, you will always eventually find more examples of twins. Nevertheless, - and this may come as a surprise to you − it is not known whether this is in fact true. Possibly they come to an end. But it seems more likely that − like the primes − the sequence of prime twins goes on forever. However, Mathematics has yet to provide a rigorous proof.
One of the things mathematicians do when they don't understand something is produce bigger and better examples of the objects that are puzzling them. We run out of ideas, so we gather more data − and this is just what we are doing at this site; if you look ahead to section 2, you will see that I have collected together the ten largest known prime twins.
Prime Triplets
If you search the list for triples of primes {p, p + 2, p + 4}, you will not find very many. In fact there is only one, {3, 5, 7}, right at the beginning. And it's easy to see why. 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.
Prime Quadruplets
Similar considerations apply to groups of four, where this time we require each of {p, p + 2, p + 6, p + 8} to be prime. Once again, it looks as if they go on indefinitely. The smallest is {5, 7, 11, 13}. We don't count {2, 3, 5, 7} even though it is a denser grouping. It is an isolated example which doesn't fit into the scheme of things. Nor, for more technical reasons, do we count {3, 5, 7, 11}.
The sequence continues with {11, 13, 17, 19}, {101, 103, 107, 109}, {191, 193, 197, 199}, {821, 823, 827, 829}, .... The usual name is prime quadruplets, although I have also seen the terms full house, inter-decal prime quartet (!) and prime decade − a reference to the pattern made by their decimal digits. 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 proven prime quadruplet of 2000 digits, found by Gerd Lamprecht in Oct 2017:
Prime k-tuplets
We can go on to define prime quintuplets, sextuplets, septuplets, octuplets, nonuplets, and so on. I had to go to the full Oxford English Dictionary for the last one − the Concise Oxford jumps from 'octuplets' to 'decuplets'. The OED also defines 'dodecuplets', but apparently there are no words for any of the others. Presumably I could make them up, but instead I shall use the number itself when I want to refer to, for example, prime 11-tuplets. I couldn't find the general term 'k-tuplets' in the OED either, but it is the word that seems to be in common use by the mathematical community.
For now, I will define a prime k-tuplet as a sequence of consecutive prime numbers such that the distance between the first and the last is in some sense as small as possible. If you think I am being too vague, there is a more precise definition later on.
At this site I have collected together what I believe to be the largest known prime k-tuplets for k = 2, 3, 4, ..., 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.
Notation
Multiplication is often denoted by an asterisk: x•y is x times y.
For k > 2, the somewhat bizarre notation N + b1, b2, ..., bk is used (only in linked pages) to denote the k numbers {N + b1, N + b2, ..., N + bk}.
Prime twins are represented as N ± 1, which is short for N plus one and N minus one.
I also use the notation n# of Caldwell and Dubner [CD93] as a convenient shorthand for the product of all the primes less than or equal to n. Thus, for example, 20# = 2•3•5•7•11•13•17•19 = 9699690.
Finally ...
I would like to keep this site as up to date as possible. Therefore, can I urge you to please send any new, large prime k-tuplets to me. You can see what I mean by 'large' by studying the lists. If the numbers are not too big, say up to 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: pzktupel[at]pzktupel[dot]de.
| Overview of largest known & early discovery of a non-trivial prime k-tuplet to given pattern. |
| Prime Counting Functions • Tables of values of π(x) up to π21(x) |
| Tables of values of πk(10n) n=1..17, k=1..16 |
| Record Gap Tables of prime k-tuplets / First Occurrence Gap |
| Initial members of prime k-tuplets |
| The smallest n-digit prime k-tuplets |
| First initial members of consecutive prime k-tuplets (PART I) |
| First initial members of consecutive prime k-tuplets (PART II) |
| Possible patterns & the Hardy-Littlewood constants of prime k-tuplets [HL22]. |
| Rank | The Largest Known Twin Primes | Digits | When | Additions |
| 1 | 2996863034895 • 21290000 ± 1 | 388342 | 19 Sep 2016 | Tom Greer, TwinGen, PrimeGrid, LLR |
| 2 | 3756801695685 • 2666669 ± 1 | 200700 | 26 Dec 2011 | Timothy Winslow, TwinGen, PrimeGrid, LLR |
| 3 | 65516468355 • 2333333 ± 1 | 100355 | 15 Aug 2009 | Peter Kaiser, NewPGen, PrimeGrid, TPS, LLR |
| 4 | 160204065 • 2262148 ± 1 | 78923 | 8 Jul 2021 | Erwin Doescher, LLR |
| 5 | 12770275971 • 2222225 ± 1 | 66907 | 4 Jul 2017 | Bo Tornberg, TwinGen, LLR |
| 6 | 12599682117 • 2211088 ± 1 | 63554 | 22 Feb 2022 | Michael Kwok, PSieve, LLR |
| 7 | 12566577633 • 2211088 ± 1 | 63554 | 22 Feb 2022 | Michael Kwok, PSieve, LLR |
| 8 | 70965694293 • 2200006 ± 1 | 60219 | 2 Apr 2016 | S. Urushihata |
| 9 | 66444866235 • 2200003 ± 1 | 60218 | 2 Apr 2016 | S. Urushihata |
| 10 | 4884940623 • 2198800 ± 1 | 59855 | 3 Jul 2015 | Michael Kwok, PSieve, LLR |
| More Twin Primes | ||||
| Rank | The Largest Known Primes Triplets | Digits | When | Additions | Certificates |
| 1 | 4111286921397 • 266420 − 1 + d, d = 0, 2, 6 | 20008 | 24 Apr 2019 | Peter Kaiser, Polysieve, LLR, Primo | click |
| 2 | 6521953289619 • 255555 − 5 + d, d = 0, 4, 6 | 16737 | 30 Apr 2013 | Peter Kaiser | click |
| 3 | 56667641271 • 244441 − 1 + d, d = 0, 2, 6 | 13389 | 1 Apr 2022 | Stephan Schöler, NewPGen, OpenPFGW; Oliver Kruse, Primo | click |
| 4 | 4207993863 • 238624 − 1 + d, d = 0, 2, 6 | 11637 | 5 Jun 2021 | Frank Doornink, NewPGen, LLR, Primo | click |
| 5 | 14059969053 • 236672 − 5 + d, d = 0, 4, 6 | 11050 | 17 Jun 2018 | Serge Batalov, NewPGen, OpenPFGW, Primo | click |
| 6 | 3221449497221499 • 234567 − 1 + d, d = 0, 2, 6 | 10422 | 2 Sep 2015 | Peter Kaiser, NewPGen, LLR, OpenPFGW, Primo | click |
| 7 | 1288726869465789 • 234567 − 5 + d, d = 0, 4, 6 | 10421 | 23 Apr 2014 | Peter Kaiser, Primo | click |
| 8 | 647935598824239 • 233619 − 1 + d, d = 0, 2, 6 | 10136 | 22 May 2019 | Peter Kaiser, Primo | click |
| 9 | 209102639346537 • 233620 − 1 + d, d = 0, 2, 6 | 10135 | 22 May 2019 | Peter Kaiser, Primo | click |
| 10 | 185353103135997 • 233620 − 1 + d, d = 0, 2, 6 | 10135 | 22 May 2019 | Peter Kaiser, Primo | click |
| More Prime Triplets | |||||
| Rank | The Largest Known Prime Quadruplets | Digits | When | Additions | Certificates |
| 1 | 667674063382677 • 233608 − 1 + d, d = 0, 2, 6, 8 | 10132 | 27 Feb 2019 | Peter Kaiser, Primo | click |
| 2 | 4122429552750669 • 216567 − 1 + d, d = 0, 2, 6, 8 | 5003 | 10 Mar 2016 | Peter Kaiser, GSIEVE, NewPGen, LLR, Primo | click |
| 3 | (1049713153083 • 2917# • (567 • 2917# + 1) + 11#) • (567 • 2917# − 1) / 7# + 1 + d, d = 0, 2, 6, 8 | 3753 | 22 Jul 2023 | Ken Davis, APTreeSieve, OpenPFGW, Primo | click |
| 4 | 101406820312263 • 212042 − 1 + d, d = 0, 2, 6, 8 | 3640 | 13 Jun 2018 | Serge Batalov, OpenPFGW, NewPGen, Primo | click |
| 5 | 2673092556681 • 153048 − 4 + d, d = 0, 2, 6, 8 | 3598 | 14 Sep 2015 | Serge Batalov, OpenPFGW, NewPGen, Primo | click |
| 6 | 2339662057597 • 103490 + 1 + d, d = 0, 2, 6, 8 | 3503 | 21 Dec 2013 | Serge Batalov, OpenPFGW, NewPGen, Primo | click |
| 7 | 305136484659 • 211399 − 1 + d, d = 0, 2, 6, 8 | 3443 | 28 Sep 2013 | Serge Batalov, OpenPFGW, NewPGen, Primo | click |
| 8 | 722047383902589 • 211111 − 1 + d, d = 0, 2, 6, 8 | 3360 | 20 Apr 2013 | Reto Keiser, NewPGen, OpenPFGW, Primo | click |
| 9 | 585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8 | 3344 | 6 Mar 2022 | Peter Kaiser, OpenPFGW, Primo | click |
| 10 | 43697976428649 • 29999 − 1 + d, d = 0, 2, 6, 8 | 3024 | 24 Mar 2012 | Peter Kaiser, Primo 3.0.9 | click |
| More Prime Quadruplets | |||||
| Rank | The Largest Known Prime Quintuplets | Digits | When | Additions | Certificates |
| 1 | 585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8, 12 | 3344 | 6 Mar 2022 | Peter Kaiser, OpenPFGW, Primo | click |
| 2 | 566761969187 • 4733# / 2 − 8 + d, d = 0, 4, 6, 10, 12 | 2034 | 6 Dec 2020 | Serge Batalov, NewPGen, OpenPFGW, Primo | click |
| 3 | 126831252923413 • 4657# / 273 + 1 + d, d = 0, 2, 6, 8, 12 | 2002 | 8 Nov 2020 | Peter Kaiser, Primo | click |
| 4 | 394254311495 • 3733# / 2 − 8 + d, d = 0, 4, 6, 10, 12 | 1606 | 30 Nov 2017 | Serge Batalov, NewPGen, OpenPFGW, Primo | click |
| 5 | 2316765173284 • 3600# + 16061 + d, d = 0, 2, 6, 8, 12 | 1543 | 16 Oct 2016 | Norman Luhn, Primo | click |
| 6 | 163252711105 • 3371# / 2 − 8 + d, d = 0, 4, 6, 10, 12 | 1443 | 1 Jan 2014 | Serge Batalov, OpenPFGW, NewPGen, Primo | click |
| 7 | 9039840848561 • 3299# / 35 − 5 + d, d = 0, 4, 6, 10, 12 | 1401 | 28 Dec 2013 | Serge Batalov, OpenPFGW, NewPGen, Primo | click |
| 8 | 699549860111847 • 24244 − 1 + d, d = 0, 2, 6, 8, 12 | 1293 | 3 Dec 2013 | Reto Keiser, R. Gerbicz, OpenPFGW, Primo | click |
| 9 | 101199 + 20483870459152351 + d, d = 0, 2, 6, 8, 12 | 1200 | 3 Mar 2023 | Norman Luhn, OpenPFGW, Primo 3.0.9 | click |
| 10 | 101199 + 7033048489975137 + d, d = 0, 4, 6, 10, 12 | 1200 | 17 Mar 2023 | Norman Luhn, OpenPFGW, Primo 3.0.9 | click |
| More Prime Quintuplets | |||||
| Rank | The Largest Known Prime Sextuplets | Digits | When | Additions | Certificates |
| 1 | 23700 + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16 | 1114 | 8 Nov 2021 | Vidar Nakling, Primo, Sixfinder ( based on Riecoin miners ) |
click |
| 2 | 28993093368077 • 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16 | 1037 | 14 Mar 2016 | Norman Luhn, APSIEVE, Primo | click |
| 3 | 6646873760397777881866826327962099685830865900246688640856 • 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16 | 780 | 8 Nov 2018 | Vidar Nakling, Primo | - |
| 4 | 29720510172503062360713760607985203309940766118866743491802189150471978534404249 • 22299 + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16 |
772 | 28 Jan 2018 | Riecoin #822096 | - |
| 5 | 29749903422302373222996698880833194129159047179535887991184960156219652236318921 • 22293 + 679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16 |
770 | 28 Sep 2017 | Riecoin #793872 | - |
| 6 | 29696802688480280387313212926526693549449146292085717645262228449092881114972806 • 22290 + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16 |
769 | 25 Feb 2018 | Riecoin #838224 | - |
| 7 | 29744205023784420961031622414734790416939049568996819659808238403983863222665068 • 22288 + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16 |
769 | 18 Feb 2018 | Riecoin #834192 | - |
| 8 | 29707412718946949415029080194980493978605678414396606766712262274235284928962561 • 22278 + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16 |
766 | 14 Jan 2018 | Riecoin #814032 | - |
| 9 | 29696978890366869883141509418765838581871522982358338407613039711378021084519043 • 22259 + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16 |
766 | 31 Dec 2017 | Riecoin #805968 | - |
| 10 | 29691575669072177222494655186416928710256802541243921484227880404600991044790342 • 22259 + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16 |
760 | 16 Dec 2017 | Riecoin #797904 | - |
| More Prime Sextuplets | |||||
| Rank | The Largest Known Prime Septuplets | Digits | When | Additions | Certificates |
| 1 | 113225039190926127209 • 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 | 1002 | 27 Jan 2021 | Peter Kaiser | click |
| 2 | 3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473 9224962020153653925994202321897236902676229040360901005487309186655777663859063397693729163631275766 0779987530903845763711693853827939526026506444774774261236889041020217108597484837589978261046949778 7199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089 • 317# + 1068701 + d, d = 0, 2, 6, 8, 12, 18, 20 |
527 | 16 Jun 2019 | Vidar Nakling,
rieMiner 0.9, Primo |
- |
| 3 | 115828580393941 • 1200# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20 | 515 | 18 Jan 2018 | Norman Luhn, Primo | - |
| 4 | 1749454900366668261124366444904767913042 • 1051# + 235367427477641 + d, d = 0, 2, 6, 8, 12, 18, 20 | 482 | 10 Sep 2023 | Michalis Christou, rieMiner | - |
| 5 | 1749440332764626179112277073423987083854 • 1051# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20 | 482 | 10 Sep 2023 | Michalis Christou, rieMiner | - |
| 6 | 1785036139318226774335532641427159030322 • 991# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20 | 452 | 10 Sep 2023 | Michalis Christou, rieMiner | - |
| 7 | 4733578067069 • 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20 | 402 | 9 May 2016 | Norman Luhn, Primo | - |
| 8 | 362084778371560960224893228648902482898 • 863# + 235367427477641 + d, d = 0, 2, 6, 8, 12, 18, 20 | 401 | 10 Sep 2023 | Michalis Christou, rieMiner | - |
| 9 | 362084724493692554397476622571818362892 • 863# + 235290683530361 + d, d = 0, 2, 6, 8, 12, 18, 20 | 401 | 10 Sep 2023 | Michalis Christou, rieMiner | - |
| 10 | 362084689974572819018431260315189393115 • 863# + 220452326319761 + d, d = 0, 2, 6, 8, 12, 18, 20 | 401 | 10 Sep 2023 | Michalis Christou, rieMiner | - |
| More Prime Septuplets | |||||
| Rank | The Largest Known Prime Octuplets | Digits | When | Additions |
| 1 | 362079385668757696008683096558661746463 • 863# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 401 | 10 Sep 2023 | Michalis Christou, rieMiner 0.93a |
| 2 | 17823192282008874449172703428792123231110 • 771# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 362 | 12 Jan 2023 | Michalis Christou, rieMiner |
| 3 | 531258360785860208657753 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 333 | 30 Sep 2022 | Peter Kaiser, Primo |
| 4 | 530956818040688210255681 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 333 | 30 Sep 2022 | Peter Kaiser, Primo |
| 5 | 697723422149271424870176724491962624555 • 701# + 145888993435301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 332 | 28 Apr 2022 | Michalis Christou, rieMiner 0.91 |
| 6 | 5586218959960365309179 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 331 | 12 Sep 2022 | Peter Kaiser, Primo |
| 7 | 4869586684665128135306 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 331 | 30 Sep 2022 | Peter Kaiser, Primo |
| 8 | 6879356578124627875380298699944709053335 • 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 | 324 | 12 Mar 2021 | Michalis Christou, rieMiner 0.91 |
| 9 | 85942978608490853163266464829675186732716531436220205198524648761309585030760262728948076619827920 • 541# + 301570107719123 + d, d = 0, 6, 8, 14, 18, 20, 24, 26 |
318 | 7 May 2023 | Riecoin #1927251 |
| 10 | 12535667484694596950800617890716044333306550769988948320664333022424209836994030121791042799528403451 • 523# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 |
318 | 11 May 2023 | Riecoin #1929345 |
| More Prime Octuplets | ||||
| Prime Octuplets, found by Riecoin | ||||
| Rank | The Largest Known Prime Nonuplets | Digits | When | Additions |
| 1 | 182075127245948453356763852678412657384571384320476086323955359028566228121357180020362596219 • 541# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 |
312 | 22 Apr 2023 | Riecoin #1918654 |
| 2 | 6981459541055817191260362842479625063402912945070015867718881817316331990854697141515826226327285164890 • 503# + 301713410008249 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 |
312 | 7 May 2023 | Riecoin #1926945 |
| 3 | 7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341 • 503# + 220469307413891 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 |
309 | 16 Feb 2022 | Riecoin #1669866 |
| 4 | 4544802941746849322755400504979525801125583245162538446615359445516948017861576296257331705348 • 521# + 301713410008249 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 |
308 | 27 May 2023 | Riecoin #1938898 |
| 5 | 154140554027854736762047378850909920571712371864836723122728277535436255651743244125969075784222257 • 503# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 |
307 | 27 Apr 2023 | Riecoin #1921294 |
| 6 | 8096703388538745581027194751126786700937232194317894651023717668314141648189594312628024001432 • 509# + 145844141558231 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 |
306 | 14 Sep 2023 | Riecoin #2002325 |
| 7 | 785736482192035162210293617854084866892633779128067177293795495786556268126995707998914641524 • 509# + 301856712297379 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 |
305 | 20 Aug 2023 | Riecoin #1987876 |
| 8 | 121133517275115547018555742318180482744008312012716527913198477819170362242224836572439329489655 • 503# + 226554621544609 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 |
304 | 12 Jun 2023 | Riecoin #1947664 | 9 | 8765572177843184814229154326623059563840136253586624568046789391874155387479145540897589092413205411 • 491# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 |
304 | 29 Jun 2023 | Riecoin #1957788 |
| 10 | 899633485111333009273839308948792075378578512165122817726236874052643987907999568251525668195739438 • 491# + 443216221842829 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30 |
303 | 17 May 2023 | Riecoin #1932821 |
| More Prime Nonuplets | ||||
| Prime Nonuplets, found by Riecoin | ||||
| Rank | The Largest Known Prime 10-tuplets | Digits | When | Additions | |
| 1 | 14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550 • 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 |
282 | 12 Sep 2021 | Riecoin #1579367 | |
| 2 | 290901656335108169864195656135043662615199446375386143995339722400236057821426952579658098504166333411889 • 401# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 |
269 | 27 Jul 2021 | Riecoin #1551825 | |
| 3 | 14257429881902877844339877915045298096140599288873476083093543949692946630381247693511330479634493 • 367# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 |
246 | 18 Aug 2022 | Riecoin #1775788 | |
| 4 | 33521646378383216495527 • 331# + 4700094892301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 | 156 | 4 Apr 2020 | Thomas Nguyen, rieMiner 0.91, MPZ-APRCL |
|
| 5 | 772556746441918 • 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 | 136 | 9 Feb 2017 | Norman Luhn | |
| 6 | 7425 • 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 | 120 | 27 May 2016 | Roger Thompson | |
| 7 | 118557188915212 • 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 | 118 | 27 Jun 2014 | Norman Luhn | |
| 8 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 53586844409797545 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 9 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 51143234991402697 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 10 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 50679161987995696 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| More Prime 10-tuplets | |||||
| Rank | The Largest Known Prime 11-tuplets | Digits | When | Additions | |
| 1 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 49376500222690335 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 2 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 46622982649030457 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 3 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 30796489110940369 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 4 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 5 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20731977215353082 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 6 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 20118509988610513 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 7 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 15866045335517629 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 8 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 5238271627884665 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
107 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 9 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 4471872451082759 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
107 | 28 May 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 10 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 1296173254392493 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36 |
107 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| More Prime 11-tuplets | |||||
| Rank | The Largest Known Prime 12-tuplets | Digits | When | Additions | |
| 1 | 13243795731372733191902494675154142263612189966992593522251560981597803197621024152571147501 + 27407861785763183 • 229# + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 |
108 | 23 Sep 2019 | Peter Kaiser, David Stevens, Polysieve, OpenPFGW, Primo |
|
| 2 | 613176722801194 • 151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 | 75 | 30 Sep 2014 | Michael Stocker, Primo |
|
| 3 | 467756 • 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | 66 | 20 May 2014 | Roger Thompson |
|
| 4 | 9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | 66 | 1 Oct 2021 | Roger Thompson |
|
| 5 | 9985397181 • 139# + 249386599747880711 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | 66 | 1 Oct 2021 | Roger Thompson |
|
| 6 | 59125383480754 • 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 | 61 | 9 Sep 2013 | Michael Stocker |
|
| 7 | 78989413043158 • 109# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | 59 | 18 Jan 2010 | Norman Luhn |
|
| 8 | 450725899 • 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | 56 | 4 Nov 2014 | Martin Raab |
|
| 9 | 1000000000000000000000000000000002955087732304487826931 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42 | 55 | 22 Sep 2022 | Norman Luhn |
|
| 10 | 1000000000000000000000000000000002760339313453283246757 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42 | 55 | 4 Oct 2022 | Norman Luhn |
|
| More Prime 12-tuplets | |||||
| Rank | The Largest Known Prime 13-tuplets | Digits | When | Additions | |
| 1 | 9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 | 66 | 1 Oct 2021 | Roger Thompson | |
| 2 | 4135997219394611 • 109# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 | 61 | 23 Mar 2017 | Norman Luhn |
|
| 3 | 14815550 • 107# + 4385574275277313 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48 | 50 | 5 Feb 2013 | Roger Thompson |
|
| 4 | 14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 | 50 | 5 Feb 2013 | Roger Thompson |
|
| 5 | 10000000000000000000000000019294427203099948114321 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 | 50 | 7 May 2023 | Norman Luhn |
|
| 6 | 61571 • 107# + 4803194122972361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 | 48 | 7 Aug 2009 | Jens Kruse Andersen |
|
| 7 | 381955327397348 • 79# + 18393211 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48 | 46 | 28 Dec 2007 | Norman Luhn |
|
| 8 | 381955327397348 • 79# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 | 46 | 28 Dec 2007 | Norman Luhn |
|
| 9 | 100000000000000000000006149198224095343810309 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48 | 45 | 2 Mar 2022 | Norman Luhn |
|
| 10 | 100000000000000000000004356680452416578030761 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48 | 45 | 5 Feb 2022 | Norman Luhn |
|
| More Prime 13-tuplets | |||||
| Rank | The Largest Known Prime 14-tuplets | Digits | When | Additions | |
| 1 | 14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | 50 | 5 Feb 2013 | Roger Thompson | |
| 2 | 381955327397348 • 79# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 | 46 | 28 Dec 2007 | Norman Luhn |
|
| 3 | 1000000000000000014210159036148101380471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | 40 | 10 Mar 2021 | Norman Luhn |
|
| 4 | 1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 | 40 | 10 Mar 2021 | Norman Luhn |
|
| 5 | 10000000000009283441665311798539399 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50 | 35 | 18 Feb 2021 | Norman Luhn |
|
| 6 | 10000000000001275924044876917671361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | 35 | 18 Feb 2021 | Norman Luhn |
|
| 7 | 26093748 • 67# + 383123187762431 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | 33 | 8 Feb 2005 | Christ van Willegen & Jens Kruse Andersen |
|
| 8 | 108804167016152508211944400342691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | 33 | 14 Apr 2008 | Jens Kruse Andersen |
|
| 9 | 107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | 33 | 14 Apr 2008 | Jens Kruse Andersen |
|
| 10 | 101885197790002105359911556070661 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50 | 33 | 14 Apr 2008 | Jens Kruse Andersen |
|
| More Prime 14-tuplets | |||||
| Rank | The Largest Known Prime 15-tuplets | Digits | When | Additions | |
| 1 | 33554294028531569 • 61# + 57800747 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 | 40 | 25 Jan 2017 | Norman Luhn | |
| 2 | 322255 • 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 | 35 | 18 Nov 2016 | Roger Thompson | |
| 3 | 10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 | 35 | 4 Sep 2012 | Roger Thompson | |
| 4 | 94 • 79# + 1341680294611244014367 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 | 33 | 5 Feb 2021 | Roger Thompson | |
| 5 | 3684 • 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56 | 33 | 5 Feb 2021 | Roger Thompson | |
| 6 | 107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 | 33 | 14 Apr 2008 | Jens Kruse Andersen | |
| 7 | 99999999948164978600250563546411 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 | 32 | 29 Nov 2004 | Jörg Waldvogel and Peter Leikauf | |
| 8 | 1251030012595955901312188450381 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 | 31 | 16 Oct 2003 | Hans Rosenthal & Jens Kruse Andersen | |
| 9 | 1100916249233879857334075234831 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50, 56 | 31 | 16 Oct 2003 | Hans Rosenthal & Jens Kruse Andersen | |
| 10 | 1003234871202624616703163933857 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56 | 31 | 9 Aug 2012 | Roger Thompson | |
| More Prime 15-tuplets | |||||
| Rank | The Largest Known Prime 16-tuplets | Digits | When | Additions | |
| 1 | 322255 • 73# + 1354238543317302647 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | 35 | 18 Nov 2016 | Roger Thompson | |
| 2 | 94 • 79# + 1341680294611244014363 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 | 33 | 5 Feb 2021 | Roger Thompson | |
| 3 | 3684 • 73# + 880858118723497737827 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | 33 | 5 Feb 2021 | Roger Thompson | |
| 4 | 1003234871202624616703163933853 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 | 31 | 9 Aug 2012 | Roger Thompson | |
| 5 | 11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60 | 29 | 2 Jan 2012 | Roger Thompson | |
| 6 | 5867208169546174917450988007 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | 28 | 11 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 7 | 5621078036155517013724659017 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | 28 | 4 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 8 | 4668263977931056970475231227 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 9 | 4652363394518920290108071177 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 10 | 4483200447126419500533043997 + d, d = 0, 2, 6, 12, 14, 20, 26, 30, 32, 36, 42, 44, 50, 54, 56, 60 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| More Prime 16-tuplets | |||||
| Rank | The Largest Known Prime 17-tuplets | Digits | When | Additions | |
| 1 | 3684 • 73# + 880858118723497737821 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | 33 | 5 Feb 2021 | Roger Thompson | |
| 2 | 100845391935878564991556707107 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 | 30 | 19 Feb 2013 | Roger Thompson | |
| 3 | 11413975438568556104209245223 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48, 54, 58, 60, 66 | 29 | 2 Jan 2012 | Roger Thompson | |
| 4 | 11410793439953412180643704677 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56, 62, 66 | 29 | 2 Jan 2012 | Roger Thompson | |
| 5 | 5867208169546174917450988001 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | 28 | 11 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 6 | 5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 | 28 | 11 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 7 | 5621078036155517013724659011 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | 28 | 4 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 8 | 5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 | 28 | 4 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 9 | 4668263977931056970475231221 + d, d = 0, 6, 8, 12, 18, 20, 26, 32, 36, 38, 42, 48, 50, 56, 60, 62, 66 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 10 | 4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| More Prime 17-tuplets | |||||
| Rank | The Largest Known Prime 18-tuplets | Digits | When | Additions | |
| 1 | 5867208169546174917450987997 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 11 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 2 | 5621078036155517013724659007 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 4 Mar 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 3 | 4668263977931056970475231217 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 4 | 4652363394518920290108071167 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 5 | 4483200447126419500533043987 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 4 Jan 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 6 | 3361885098594416802447362317 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 30 Jul 2013 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 7 | 3261917553005305074228431077 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 30 Jul 2013 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 8 | 3176488693054534709318830357 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 30 Jul 2013 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 9 | 2650778861583720495199114537 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 25 Feb 2013 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 10 | 2406179998282157386567481197 + d, d = 0, 4, 10, 12, 16, 22, 24, 30, 36, 40, 42, 46, 52, 54, 60, 64, 66, 70 | 28 | 31 Dec 2012 | Raanan Chermoni & Jaroslaw Wroblewski | |
| More Prime 18-tuplets | |||||
| Rank | The Largest Known Prime 19-tuplets | Digits | When | Additions | 1 | 622803914376064301858782434517 + d, d = 0, 4, 6, 10, 12, 16, 24, 30, 34, 40, 42, 46, 52, 54, 60, 66, 70, 72, 76 | 30 | 27 Dec 2018 | Raanan Chermoni & Jaroslaw Wroblewski |
| 2 | 248283957683772055928836513597 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76 | 30 | 1 Aug 2016 | Raanan Chermoni & Jaroslaw Wroblewski |
| 3 | 138433730977092118055599751677 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76 | 30 | 8 Oct 2015 | Raanan Chermoni & Jaroslaw Wroblewski |
| 4 | 39433867730216371575457664407 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76 | 29 | 8 Jan 2015 | Raanan Chermoni & Jaroslaw Wroblewski |
| 5 | 2406179998282157386567481191 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 | 28 | 31 Dec 2012 | Raanan Chermoni & Jaroslaw Wroblewski |
| 6 | 2348190884512663974906615481 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 | 28 | 17 Dec 2012 | Raanan Chermoni & Jaroslaw Wroblewski |
| 7 | 917810189564189435979968491 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 | 27 | 29 May 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 8 | 656632460108426841186109951 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 | 27 | 19 Feb 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 9 | 630134041802574490482213901 + d, d = 0, 6, 10, 16, 18, 22, 28, 30, 36, 42, 46, 48, 52, 58, 60, 66, 70, 72, 76 | 27 | 9 Feb 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 10 | 37 + d, d = 0, 4, 6, 10, 16, 22, 24, 30, 34, 36, 42, 46, 52, 60, 64, 66, 70, 72, 76 | 2-3 | - | - |
| More Prime 19-tuplets | ||||
| Rank | The Largest Known Prime 20-tuplets | Digits | When | Additions | 1 | 1236637204227022808686214288579 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | 31 | 23 May 2021 | Raanan Chermoni & Jaroslaw Wroblewski |
| 2 | 1188350591359110800209379560799 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | 31 | 21 Jan 2021 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 3 | 1153897621507935436463788957529 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | 31 | 26 Dec 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 4 | 1135540756371356698957890225821 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 | 31 | 19 Dec 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 5 | 1126002593922465663847897293731 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 | 31 | 17 Nov 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 6 | 1094372814043722195189448411199 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | 31 | 20 Oct 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 7 | 1060475118776959297139870952701 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 | 31 | 18 Sep 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 8 | 999627565307688186459783232931 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 | 30 | 19 Jun 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 9 | 957278727962618711849051282459 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80 | 30 | 23 Mar 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 10 | 839013472011818416634745523991 + d, d = 0, 2, 6, 8, 12, 20, 26, 30, 36, 38, 42, 48, 50, 56, 62, 66, 68, 72, 78, 80 | 30 | 28 Oct 2020 | Raanan Chermoni & Jaroslaw Wroblewski | |
| More Prime 20-tuplets | |||||
| Rank | The Largest Known Prime 21-tuplets | Digits | When | Additions | 1 | 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 | 27 Dec 2018 | Raanan Chermoni & Jaroslaw Wroblewski |
| 2 | 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 | 1 Aug 2016 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 3 | 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 | 8 Oct 2015 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 4 | 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 | 8 Jan 2015 | Raanan Chermoni & Jaroslaw Wroblewski | |
| 5 | 29 + d, d = 0, 2, 8, 12, 14, 18, 24, 30, 32, 38, 42, 44, 50, 54, 60, 68, 72, 74, 78, 80, 84 | 2-3 | - | - | |
| More Prime 21-tuplets | |||||
A prime k-tuplet is a sequence of k consecutive prime numbers such that in some sense the difference between the first and the last is as small as possible. The idea is to generalise the concept of prime twins.
More precisely: We first define s(k) to be the smallest number s for which there exist k integers b1 < b2 < ... < bk, bk − b1 = s and, for every prime q, not all the residues modulo q are represented by b1, b2, ..., bk. A prime k-tuplet is then defined as a sequence of consecutive primes {p1, p2, ..., pk} such that for every prime q, not all the residues modulo q are represented by p1, p2, ..., pk, pk − p1 = s(k). Observe that the definition might exclude a finite number (for each k) of dense clusters at the beginning of the prime number sequence − for example, {97, 101, 103, 107, 109} satisfies the conditions of the definition of a prime 5-tuplet , but {3, 5, 7, 11, 13} doesn't because all three residues modulo 3 are represented.
Patterns of Prime k-tuplets
The simplest case is s(2) = 2, corresponding to prime twins: {p, p + 2}. Next, s(3) = 6 and two types of prime triplets: {p, p + 2, p + 6} and {p, p + 4, p + 6}, followed by s(4) = 8 with just one pattern: {p, p + 2, p + 6, p + 8} of prime quadruplets. The sequence continues with s(5) = 12, s(6) = 16, s(7) = 20, s(8) = 26, s(9) = 30, s(10) = 32, s(11) = 36, s(12) = 42, s(13) = 48, s(14) = 50, s(15) = 56, s(16) = 60, s(17) = 66 and so on. It is number A008407 in N.J.A. Sloane's On-line Encyclopedia of Integer Sequences.
Primality Proving
In keeping with similar published lists, I have decided not to accept anything other than true, proven primes. Numbers which have merely passed the Fermat test, aN = a (mod N), will need to be validated. If N − 1 or N + 1 is sufficiently factorized (usually just under a third), the methods of Brillhart, Lehmer and Selfridge [BLS75] will suffice. Otherwise the numbers may have to be subjected to a general primality test, such as the Jacobi sum test of Adleman, Pomerance, Rumely, Cohen and Lenstra (APRT-CLE in UBASIC, for example), or one of the elliptic curve primality proving programs: Atkin and Morain's ECPP, or its successor, Franke, Kleinjung, Wirth and Morain's FAST-ECPP, or Marcel Martin's Primo.
Primes
Euclid proved that there are infinitely many primes. Paulo Ribenboim [Rib95] has collected together a considerable number of different proofs of this important theorem. My favourite (which is not in Ribenboim's book) goes like this: We have
∏p prime 1/(1 − 1/p2) = ∑n = 1 to ∞ 1/n2 = π2/6.
But π2 is irrational; so the product on the left cannot have a finite number of factors.
In its simplest form, the prime number theorem states that the number of primes less than x is asymptotic to x/(log x). This was first proved by Hadamard and independently by de la Vallee Poussin in 1896. Later, de la Vallee Poussin found a better estimate:
∫u = 0 to x du/(log u) + error term,
where the error term is bounded above by A x exp(−B √(log x)) for some constants A and B. With more work (H.-E. Richert, 1967), √(log x) in this last expression can be replaced by (log x)3/5(log log x)−1/5. The most important unsolved conjecture of prime number theory, indeed, all of mathematics - the Riemann Hypothesis − asserts that the error term can be bounded by a function of the form A √x log x.
The Twin Prime Conjecture
G.H. Hardy & J.E. Littlewood did the first serious work on the distribution of prime twins. In their paper 'Some problems of Partitio Numerorum: III...' [HL22], they conjectured a formula for the number of twins between 1 and x:
2 C2 x / (log x)2,
where C2 = ∏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 Hardy-Littlewood Prime k-tuple Conjecture
The Partitio Numerorum: III paper [HL22] goes on to formulate a general conjecture concerning the distribution of arbitrary groups of prime numbers (The k-tuplets of this site are special cases): Let b1, b2, ..., bk be k distinct integers. Then the number of groups of primes N + b1, N + b2, ..., N + bk between 2 and x is approximately
Hk Ck ∫u = 2 to x du / (log u)k,
where
Hk = ∏p prime, p ≤ k pk − 1 (p − v) / (p − 1)k ∏p prime, p > k, p|D (p − v) / (p − k),
Ck = ∏p prime, p > k pk − 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 |bi − bj|, 1 ≤ i < j ≤ k.
The first product in Hk is over the primes not greater than k, the second is over the primes greater than k which divide D and the product Ck 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, H2 = 2, and Ck = C2, the twin prime constant given above.
It is worth pointing out that with modern mathematical software the prime k-tuplet constants Ck 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 Ck = − ∑n = 2 to ∞ log [ζ(n) ∏p prime, p ≤ k (1 − 1/pn)] / n ∑d|n μ(n/d) (kd − k).
[BLS75] John Brillhart, D.H. Lehmer & J.L. Selfridge, New primality criteria and factorizations of 2m ± 1, Math. Comp. 29 (1975), 620-647.
[CD93] C.K.Caldwell & H. Dubner, Primorial, factorial and multifactorial primes, Math. Spectrum 26 (1993/94), 1-7.
[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
[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.