Prime k-tuple

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 additions
History of additions → 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022

Introduction

Summary

•Largest known
•Overview of largest known & early discovery of non-trivials to given pattern
•Early discoveries ( 100 up to 100000+ digits )
•Smallest prime k-tuplets ( 5 up to 20000 digits )
•Prime counting functions / Table of values
•Record Gap Tables / First Occurrence Gap (new)
•Initial members
•Patterns & Hardy-Littlewood constants

Mathematical Background

References

Useful links

If you like, you can help to save this huge work for future − please download and store it ! ( Filesize ~4.4 MB, 12 March 2023 )

1. Introduction

Prime Numbers

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

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

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

Prime Twins

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

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

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

Prime Triplets

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

10^¹⁹⁹⁹ + 205076414983951,
10^¹⁹⁹⁹ + 205076414983953,
10^¹⁹⁹⁹ + 205076414983957,
10^¹⁹⁹⁹ + 205076414983959.

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, ..., b_k is used (only in linked pages) to denote the k numbers {N + b₁, N + b₂, ..., 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.

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.

2. Summary

Overview of largest known & early discovery of a non-trivial prime k-tuplet to given pattern.

Early discovery with at least 100 to 900 digits in step of 100.

Early discovery with at least 1000 to 9000 digits in step of 1000.

Early discovery with at least 10000 to 90000 digits in step of 10000.

Early discovery with at least 100000 and more digits.

Smallest prime k-tuplets with 5 to 200 digits in step of 5.

Smallest prime k-tuplets with 100 to 2000 digits in step of 100.

Smallest prime k-tuplets with 1000 to 9000 digits in step of 1000.

Smallest prime k-tuplets with 10000 and 20000 digits.

Smallest googol prime k-tuplets.

Prime Counting Functions • Tables of values of π(x) up to π₂₁(x)

Tables of values of π_k(10^ⁿ) 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].

The Largest Known Twin Primes	Digits	When	Additions
2996863034895 • 2^1290000 ± 1	388342	19 Sep 2016	Tom Greer, TwinGen, PrimeGrid, LLR
3756801695685 • 2⁶⁶⁶⁶⁶⁹ ± 1	200700	26 Dec 2011	Timothy Winslow, TwinGen, PrimeGrid, LLR
65516468355 • 2³³³³³³ ± 1	100355	15 Aug 2009	Peter Kaiser, NewPGen, PrimeGrid, TPS, LLR
160204065 • 2²⁶²¹⁴⁸ ± 1	78923	8 Jul 2021	Erwin Doescher, LLR
12770275971 • 2²²²²²⁵ ± 1	66907	4 Jul 2017	Bo Tornberg, TwinGen, LLR
12599682117 • 2²¹¹⁰⁸⁸ ± 1	63554	22 Feb 2022	Michael Kwok, PSieve, LLR
12566577633 • 2²¹¹⁰⁸⁸ ± 1	63554	22 Feb 2022	Michael Kwok, PSieve, LLR
70965694293 • 2²⁰⁰⁰⁰⁶ ± 1	60219	2 Apr 2016	S. Urushihata
66444866235 • 2²⁰⁰⁰⁰³ ± 1	60218	2 Apr 2016	S. Urushihata
4884940623 • 2¹⁹⁸⁸⁰⁰ ± 1	59855	3 Jul 2015	Michael Kwok, PSieve, LLR
More Twin Primes

The Largest Known Prime Triplets	Digits	When	Additions	Certificates
4111286921397 • 2⁶⁶⁴²⁰ − 1 + d, d = 0, 2, 6	20008	24 Apr 2019	Peter Kaiser, Polysieve, LLR, Primo	click
6521953289619 • 2⁵⁵⁵⁵⁵ − 5 + d, d = 0, 4, 6	16737	30 Apr 2013	Peter Kaiser	click
56667641271 • 2⁴⁴⁴⁴¹ − 1 + d, d = 0, 2, 6	13389	1 Apr 2022	Stephan Schöler, NewPGen, OpenPFGW; Oliver Kruse, Primo	click
4207993863 • 2³⁸⁶²⁴ − 1 + d, d = 0, 2, 6	11637	5 Jun 2021	Frank Doornink, NewPGen, LLR, Primo	click
14059969053 • 2³⁶⁶⁷² − 5 + d, d = 0, 4, 6	11050	17 Jun 2018	Serge Batalov, NewPGen, OpenPFGW, Primo	click
3221449497221499 • 2³⁴⁵⁶⁷ − 1 + d, d = 0, 2, 6	10422	2 Sep 2015	Peter Kaiser, NewPGen, LLR, OpenPFGW, Primo	click
1288726869465789 • 2³⁴⁵⁶⁷ − 5 + d, d = 0, 4, 6	10421	23 Apr 2014	Peter Kaiser, Primo	click
647935598824239 • 2³³⁶¹⁹ − 1 + d, d = 0, 2, 6	10136	22 May 2019	Peter Kaiser, Primo	click
209102639346537 • 2³³⁶²⁰ − 1 + d, d = 0, 2, 6	10135	22 May 2019	Peter Kaiser, Primo	click
185353103135997 • 2³³⁶²⁰ − 1 + d, d = 0, 2, 6	10135	22 May 2019	Peter Kaiser, Primo	click
More Prime Triplets

The Largest Known Prime Quadruplets	Digits	When	Additions	Certificates
667674063382677 • 2³³⁶⁰⁸ − 1 + d, d = 0, 2, 6, 8	10132	27 Feb 2019	Peter Kaiser, Primo	click
4122429552750669 • 2¹⁶⁵⁶⁷ − 1 + d, d = 0, 2, 6, 8	5003	10 Mar 2016	Peter Kaiser, GSIEVE, NewPGen, LLR, Primo	click
101406820312263 • 2¹²⁰⁴² − 1 + d, d = 0, 2, 6, 8	3640	13 Jun 2018	Serge Batalov, OpenPFGW, NewPGen, Primo	click
2673092556681 • 15³⁰⁴⁸ − 4 + d, d = 0, 2, 6, 8	3598	14 Sep 2015	Serge Batalov, OpenPFGW, NewPGen, Primo	click
2339662057597 • 10³⁴⁹⁰ + 1 + d, d = 0, 2, 6, 8	3503	21 Dec 2013	Serge Batalov, OpenPFGW, NewPGen, Primo	click
305136484659 • 2¹¹³⁹⁹ − 1 + d, d = 0, 2, 6, 8	3443	28 Sep 2013	Serge Batalov, OpenPFGW, NewPGen, Primo	click
722047383902589 • 2¹¹¹¹¹ − 1 + d, d = 0, 2, 6, 8	3360	20 Apr 2013	Reto Keiser, NewPGen, OpenPFGW, Primo	click
585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8	3344	06 Mar 2022	Peter Kaiser, OpenPFGW, Primo	click
43697976428649 • 2⁹⁹⁹⁹ − 1 + d, d = 0, 2, 6, 8	3024	24 Mar 2012	Peter Kaiser, Primo 3.0.9	click
10²⁹⁹⁹ + 339930644528851 + d, d = 0, 2, 6, 8	3000	04 May 2022	Norman Luhn, NewPGen, OpenPFGW, Primo	click
More Prime Quadruplets

The Largest Known Prime Quintuplets	Digits	When	Additions	Certificates
585150568069684836 • 7757# / 85085 + 5 + d, d = 0, 2, 6, 8, 12	3344	06 Mar 2022	Peter Kaiser, OpenPFGW, Primo	click
566761969187 • 4733# / 2 − 8 + d, d = 0, 4, 6, 10, 12	2034	06 Dec 2020	Serge Batalov, NewPGen, OpenPFGW, Primo	click
126831252923413 • 4657# / 273 + 1 + d, d = 0, 2, 6, 8, 12	2002	8 Nov 2020	Peter Kaiser, Primo	click
394254311495 • 3733# / 2 − 8 + d, d = 0, 4, 6, 10, 12	1606	Nov 2017	Serge Batalov, NewPGen, OpenPFGW, Primo	click
2316765173284 • 3600# + 16061 + d, d = 0, 2, 6, 8, 12	1543	16 Oct 2016	Norman Luhn, Primo	click
163252711105 • 3371# / 2 − 8 + d, d = 0, 4, 6, 10, 12	1443	1 Jan 2014	Serge Batalov, OpenPFGW, NewPGen, Primo	click
9039840848561 • 3299# / 35 − 5 + d, d = 0, 4, 6, 10, 12	1401	Dec 2013	Serge Batalov, OpenPFGW, NewPGen, Primo	click
699549860111847 • 2⁴²⁴⁴ − 1 + d, d = 0, 2, 6, 8, 12	1293	3 Dec 2013	Reto Keiser, R. Gerbicz, OpenPFGW, Primo	click
10¹¹⁹⁹ + 20483870459152351 + d, d = 0, 2, 6, 8, 12	1200	03 Mar 2023	Norman Luhn, OpenPFGW, Primo 3.0.9	click
10¹¹⁹⁹ + 7033048489975137 + d, d = 0, 4, 6, 10, 12	1200	17 Mar 2023	Norman Luhn, OpenPFGW, Primo 3.0.9	click
More Prime Quintuplets

The Largest Known Prime Sextuplets	Digits	When	Additions	Certificates
2³⁷⁰⁰ + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16	1114	8 Nov 2021	Vidar Nakling, Primo, Sixfinder ( based on Riecoin miners )	click
28993093368077 • 2400# + 19417 + d, d = 0, 4, 6, 10, 12, 16	1037	14 Mar 2016	Norman Luhn, APSIEVE, Primo	click
6646873760397777881866826327962099685830865900246688640856 • 1699# + 43777 + d, d = 0, 4, 6, 10, 12, 16	780	8 Nov 2018	Vidar Nakling, Primo	-
29720510172503062360713760607985203309940766118866743491802189150471978534404249 • 2²²⁹⁹ + 14271253084334081637544486111223831073612730979632132919368177563415768349505 + d, d = 0, 4, 6, 10, 12, 16	772	28 Jan 2018	Riecoin #822096	-
29749903422302373222996698880833194129159047179535887991184960156219652236318921 • 2²²⁹³ + 679631792885016654160023247517239700227428004849763556497260661860592843345 + d, d = 0, 4, 6, 10, 12, 16	770	28 Sep 2017	Riecoin #793872	-
29696802688480280387313212926526693549449146292085717645262228449092881114972806 • 2²²⁹⁰ + 1946690158750077943506249776690378666457458353296002764327070450442847661633 + d, d = 0, 4, 6, 10, 12, 16	769	25 Feb 2018	Riecoin #838224	-
29744205023784420961031622414734790416939049568996819659808238403983863222665068 • 2²²⁸⁸ + 14305894933680691041378655981062938998356035914288745998258984615535179477709 + d, d = 0, 4, 6, 10, 12, 16	769	18 Feb 2018	Riecoin #834192	-
29707412718946949415029080194980493978605678414396606766712262274235284928962561 • 2²²⁷⁸ + 21774293793439586643674306888881718167342014062406478752847391700510857054773 + d, d = 0, 4, 6, 10, 12, 16	766	14 Jan 2018	Riecoin #814032	-
29696978890366869883141509418765838581871522982358338407613039711378021084519043 • 2²²⁵⁹ + 24152316155470595374357736963765392505702343434016117070743766886456802014213 + d, d = 0, 4, 6, 10, 12, 16	766	31 Dec 2017	Riecoin #805968	-
29691575669072177222494655186416928710256802541243921484227880404600991044790342 • 2²²⁵⁹ + 22953847913844494543791161053509719129919186139904030102712344430311343318911 + d, d = 0, 4, 6, 10, 12, 16	760	16 Dec 2017	Riecoin #797904	-
More Prime Sextuplets

The Largest Known Prime Septuplets	Digits	When	Additions	Certificates
113225039190926127209 • 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20	1002	27 Jan 2021	Peter Kaiser	click
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	-
115828580393941 • 1200# + 5132201 + d, d = 0, 2, 6, 8, 12, 18, 20	515	18 Jan 2018	Norman Luhn, Primo	-
4733578067069 • 940# + 626609 + d, d = 0, 2, 8, 12, 14, 18, 20	402	09 May 2016	Norman Luhn, Primo	-
17823436097302039011383530397574638170734 • 771# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20	362	12 Jan 2023	Michalis Christou, rieMiner	-
17823429667990098910404937524728999750239 • 771# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20	362	12 Jan 2023	Michalis Christou, rieMiner	-
17823427931355237791135579059273253342104 • 771# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20	362	12 Jan 2023	Michalis Christou, rieMiner	-
17823424548404400585693182916303260773687 • 771# + 145844141558231 + d, d = 0, 2, 6, 8, 12, 18, 20	362	12 Jan 2023	Michalis Christou, rieMiner	-
17823423197963241682981949738836473771455 • 771# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20	362	12 Jan 2023	Michalis Christou, rieMiner	-
17823416716191253381804284202848071984717 • 771# + 145844141558231 + d, d = 0, 2, 6, 8, 12, 18, 20	362	12 Jan 2023	Michalis Christou, rieMiner	-
More Prime Septuplets

The Largest Known Prime Octuplets	Digits	When	Additions
17823192282008874449172703428792123231110 • 771# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	362	12 Jan 2023	Michalis Christou, rieMiner
531258360785860208657753 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	333	30 Sep 2022	Peter Kaiser, Primo
530956818040688210255681 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	333	30 Sep 2022	Peter Kaiser, Primo
697723422149271424870176724491962624555 • 701# + 145888993435301 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	332	28 Apr 2022	Michalis Christou, rieMiner 0.91
5586218959960365309179 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	331	12 Sep 2022	Peter Kaiser, Primo
4869586684665128135306 • 757# / 1768767 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	331	30 Sep 2022	Peter Kaiser, Primo
6879356578124627875380298699944709053335 • 677# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	324	12 Mar 2021	Michalis Christou, rieMiner 0.91
54598824190010361875282469578684418459657573362461324471660422883073099662240278837985413217294784653805 • 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26	316	30 Oct 2021	Riecoin #1607166
237290937625019988409934680338216405908629349352492341129431599973490073614754863588338476036934867547671407908 • 487# + 1146773 + d, d = 0, 6, 8, 14, 18, 20, 24, 26	312	20 Oct 2021	Riecoin #1600958
188273324392097141944873869557423547058811920840483304365112457383885407879644413861445197917160744 • 509# + 226374233346623 + d, d = 0, 6, 8, 14, 18, 20, 24, 26	310	20 Oct 2021	Riecoin #1600978
6068138408292784654794269848877333341123929067736255007020032491702134706361073607222476583743922495929518535 • 487# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26	310	20 Oct 2021	Riecoin #1600993
More Prime Octuplets
Prime Octuplets, found by Riecoin

The Largest Known Prime Nonuplets	Digits	When	Additions
7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341 • 503# + 220469307413891 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30	309	16 Feb 2022	Bielawski Mathematicians
3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501 • 463# + 2325810733931801 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30	302	22 Aug 2021	Riecoin #1567399
1628310212800330650680875193463302528297524858575252703732019584033146050060402902119214625070 • 491# + 437171043550511 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30	297	25 Mar 2023	Riecoin #1902419
1620259924615470570706663156278905026372754732844252658390408090245313172792664271166384219300680488342402961778 • 450# + 1487854607298791 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30	296	20 Aug 2021	Riecoin #1566093
387833514641724600357029749119397331285062620621983133723181869572568059514167753188325960698719230 • 467# + 226193845148629 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30	294	27 Oct 2021	Riecoin #1605403
352360483181346865458241271679723612698522403422825810225007396594617602134749714450854301781722077875166 • 457# + 302000014586509 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30	292	27 Dec 2021	Riecoin #1640522
40893595297845006551741048717748959451570266851095389722761855002653709793065456232477944049520841797242 • 457# + 437163765888581 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30	291	7 Sep 2021	Riecoin #1576463
1285897436414229397879253137835709328420499854296690250107786136168118236699867925624486888053153988984555053 • 443# + 27899359258009 + d, d = 0, 4, 10, 12, 18, 22, 24, 28, 30	291	19 Dec 2021	Riecoin #1636021
425637736526956247129170414648410773829792758837126862506126866133457379262600484880404875012755358905043168102 • 439# + 980125031081081 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30	291	14 Mar 2022	Riecoin #1685127
1574649584907747555706133630785765300471148459969743147116987603964781873723920925748330533139472006902 • 457# + 145799289681161 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30	290	6 Dec 2021	Riecoin #1628519
More Prime Nonuplets
Prime Nonuplets, found by Riecoin

The Largest Known Prime 10-tuplets	Digits	When	Additions
14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550 • 449# + 226554621544607 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32	282	12 Sep 2021	Riecoin #1579367
290901656335108169864195656135043662615199446375386143995339722400236057821426952579658098504166333411889 • 401# + 380284918609481 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32	269	27 Jul 2021	Riecoin #1551825
14257429881902877844339877915045298096140599288873476083093543949692946630381247693511330479634493 • 367# + 114189340938131 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32	246	18 Aug 2022	Riecoin #1775788
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
772556746441918 • 300# + 29247917 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 32	136	9 Feb 2017	Norman Luhn
7425 • 281# + 471487291717627721 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32	120	27 May 2016	Roger Thompson
118557188915212 • 260# + 25658441 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32	118	27 Jun 2014	Norman Luhn
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
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
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

The Largest Known Prime 11-tuplets	Digits	When	Additions
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
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
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
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
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
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
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
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
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
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

The Largest Known Prime 12-tuplets	Digits	When	Additions
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
613176722801194 • 151# + 177321217 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42	75	30 Sep 2014	Michael Stocker, Primo
467756 • 151# + 193828829641176461 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42	66	20 May 2014	Roger Thompson
9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42	66	1 Oct 2021	Roger Thompson
9985397181 • 139# + 249386599747880711 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42	66	1 Oct 2021	Roger Thompson
59125383480754 • 113# + 12455557957 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42	61	9 Sep 2013	Michael Stocker
78989413043158 • 109# + 38458151 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42	59	18 Jan 2010	Norman Luhn
450725899 • 113# + 1748520218561 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42	56	4 Nov 2014	Martin Raab
1000000000000000000000000000000002955087732304487826931 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42	55	22 Sep 2022	Norman Luhn
1000000000000000000000000000000002760339313453283246757 + d, d = 0, 6, 10, 12, 16, 22, 24, 30, 34, 36, 40, 42	55	04 Oct 2022	Norman Luhn
More Prime 12-tuplets

The Largest Known Prime 13-tuplets	Digits	When	Additions
9985637467 • 139# + 3629868888791261 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48	66	1 Oct 2021	Roger Thompson
4135997219394611 • 110# + 117092849 + d, d = 0, 2, 12, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48	61	23 Mar 2017	Norman Luhn
14815550 • 107# + 4385574275277313 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 40, 46, 48	50	5 Feb 2013	Roger Thompson
14815550 • 107# + 4385574275277311 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48	50	5 Feb 2013	Roger Thompson
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
381955327397348 • 80# + 18393211 + d, d = 0, 6, 12, 16, 18, 22, 28, 30, 36, 40, 42, 46, 48	46	28 Dec 2007	Norman Luhn
381955327397348 • 80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48	46	28 Dec 2007	Norman Luhn
100000000000000000000006149198224095343810309 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48	45	02 Mar 2022	Norman Luhn
100000000000000000000004356680452416578030761 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48	45	05 Feb 2022	Norman Luhn
100000000000000000000002004740564798426955633 + d, d = 0, 4, 6, 10, 16, 18, 24, 28, 30, 34, 36, 46, 48	45	06 Mar 2022	Norman Luhn
More Prime 13-tuplets

The Largest Known Prime 14-tuplets	Digits	When	Additions
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
381955327397348 • 80# + 18393209 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50	46	28 Dec 2007	Norman Luhn
1000000000000000014210159036148101380471 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50	40	10 Mar 2021	Norman Luhn
1000000000000000000349508508460276218889 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50	40	10 Mar 2021	Norman Luhn
10000000000009283441665311798539399 + d, d = 0, 2, 8, 14, 18, 20, 24, 30, 32, 38, 42, 44, 48, 50	35	18 Feb 2021	Norman Luhn
10000000000001275924044876917671361 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50	35	18 Feb 2021	Norman Luhn
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
108804167016152508211944400342691 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50	33	14 Apr 2008	Jens Kruse Andersen
107173714602413868775303366934621 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30, 32, 36, 42, 48, 50	33	14 Apr 2008	Jens Kruse Andersen
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

The Largest Known Prime 15-tuplets	Digits	When	Additions
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
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
10004646546202610858599716515809907 + d, d = 0, 2, 6, 12, 14, 20, 24, 26, 30, 36, 42, 44, 50, 54, 56	35	4 Sep 2012	Roger Thompson
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
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
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
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
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
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
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

The Largest Known Prime 16-tuplets	Digits	When	Additions
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
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
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
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
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
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
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
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
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
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

The Largest Known Prime 17-tuplets	Digits	When	Additions
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
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
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
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
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
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
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
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
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
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

The Largest Known Prime 18-tuplets	Digits	When	Additions
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
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
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
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
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
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
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
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
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
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

The Largest Known Prime 19-tuplets	Digits	When	Additions
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
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
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
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
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
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
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
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
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
{37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}
More Prime 19-tuplets

The Largest Known Prime 20-tuplets	Digits	When	Additions
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
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
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
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
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
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
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
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
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
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

The Largest Known Prime 21-tuplets	Digits	When	Additions
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
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
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
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
{29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113}	-	-	-
More Prime 21-tuplets

3. Mathematical Background

Definition

A prime k-tuplet is a sequence of k consecutive prime numbers such that in some sense the difference between the first and the last is as small as possible. The idea is to generalise the concept of prime twins.

More precisely: We first define s(k) to be the smallest number s for which there exist k integers b₁ < b₂ < ... < b_k, b_k − b₁ = s and, for every prime q, not all the residues modulo q are represented by b₁, b₂, ..., b_k. A prime k-tuplet is then defined as a sequence of consecutive primes {p₁, p₂, ..., p_k} such that for every prime q, not all the residues modulo q are represented by p₁, p₂, ..., p_k, p_k − 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, 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.

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, ..., b_k be k distinct integers. Then the number of groups of primes N + b₁, N + b₂, ..., N + b_k between 2 and x is approximately

H_k C_k ∫_{u = 2 to x} du / (log u)^k,

where

H_k = ∏_{p prime, p ≤ k} p^{k − 1} (p − v) / (p − 1)^k ∏_{p prime, p > k, p|D} (p − v) / (p − k),

C_k = ∏_{p prime, p > k} p^{k − 1} (p − k) / (p − 1)^k,

v = v(p) is the number of distinct remainders of b₁, b₂, ..., b_k modulo p and D is the product of the differences |b_i − b_j|, 1 ≤ i < j ≤ k.

The first product in H_k is over the primes not greater than k, the second is over the primes greater than k which divide D and the product C_k 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 C_k = C₂, the twin prime constant given above.

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 = − ∑_{n = 2 to ∞} log [ζ(n) ∏_{p prime, p ≤ k} (1 − 1/pⁿ)] / n ∑_d|n μ(n/d) (k^d − k).

4. References

[BLS75] John Brillhart, D.H. Lehmer & J.L. Selfridge, New primality criteria and factorizations of 2^m ± 1, Math. Comp. 29 (1975), 620-647.

[CD93] C.K.Caldwell & H. Dubner, Primorial, factorial and multifactorial primes, Math. Spectrum 26 (1993/94), 1-7.

[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.

Abstract

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