Note: This page was created by Jens Kruse Andersen (2004-2019).

Updated by Norman Luhn (since Oct 2021).

Contact: pzktupel@pzktupel.de

**News**

2022

March 06: New record: 3344-digit 5-tuplet by Peter Kaiser

February 16: New record: 309-digit 9-tuplet by Bielawski Mathematicians

History Of Additions (2004 - 2021)

Introduction

The largest known simultaneous primes

Credited programs and projects

Rules

Submissions

**Introduction**

This page records the single largest known case of k simultaneous primes for
each k, and the record history.

The idea is to show the best overall record for different types of
sets with k primes.

p# (called p primorial) is the product of all primes ≤ p, e.g. 10# = 2 · 3 · 5 · 7 = 210.

An expression with p# is often used in prime searches to avoid factors ≤ p.

k | Primes | Type | Digits | Year | Discoverer | Record History |
---|---|---|---|---|---|---|

1 | 2^{82589933}-1 |
Mersenne | 24862048 | 2018 | Patrick Laroche, GIMPS | |

2 | 2996863034895 · 2^{1290000} ± 1 | Tuplet (twin) | 388342 | 2016 | Tom Greer, PrimeGrid, TwinGen, LLR | |

3 | 1128330746865 · 2^{66439} · 2^{n} - 1, n=0..2 | CC, 1st kind | 20013 | 2020 | Dr. Michael Paridon | |

4 | 667674063382677 · 2^{33608} - 1, +1, +5, +7 | Tuplet | 10132 | 2019 | Peter Kaiser, PolySieve, LLR, Primo | |

5 | 585150568069684836 · 7757# / 85085 + 5 + d, d = 0, 2, 6, 8, 12 | Tuplet | 3344 | 2022 | Peter Kaiser, OpenPFGW, Primo | |

6 | 2^{3700} + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16 | Tuplet | 1114 | 2021 | Vidar Nakling, Primo, Sixfinder | |

7 | 113225039190926127209 · 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 | Tuplet | 1002 | 2021 | Peter Kaiser | |

8 | 2373007846680317952 · 761# · 2^{n}
+ 1,
n=0..7 | CC, 2nd kind | 338 | 2016 | Andrey Balyakin | |

9 | x100 · 503# + 220469307413891 + 0, 2, 6, 8, 12, 18, 20, 26, 30 | Tuplet | 309 | 2022 | Bielawski Mathematicians | |

10 | x98 · 449# + 226554621544607 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 | Tuplet | 282 | 2021 | Riecoin #1579367 | |

11 | 341841671431409652891648 · 311# · 2^{n}
+ 1, n=0..10 | CC, 2nd kind | 151 | 2016 | Andrey Balyakin | |

12 | 906644189971753846618980352 · 233# · 2^{n}
+ 1, n=0..11 | CC, 2nd kind | 123 | 2015 | Andrey Balyakin | |

13 | x84 · 61# · 2^{n} - 1, n=0..12 | CC, 1st kind | 108 | 2014 | Primecoin | |

14 | x82 · 47# · 2^{n}
+ 1, n=0..13 | CC, 2nd kind | 102 | 2014 | Primecoin | |

15 | 14354792166345299956567113728 · 43# · 2^{n} - 1, n=0..14 | CC, 1st kind | 47 | 2015 | Andrey Balyakin | |

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

17 | 3684 · 73# + 880858118723497737821 + 0,6,8,12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 33 | 2021 | Roger Thompson | |

18 | 658189097608811942204322720 · 2^{n}
+ 1, n=0..17 | CC, 2nd kind | 30 | 2014 | Raanan Chermoni & Jaroslaw Wroblewski | |

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

20 | 1236637204227022808686214288579 + 0,2,8,12,14,18,24,30,32,38,42,44,50,54,60,68,72,74,78,80 | Tuplet | 31 | 2021 | Raanan Chermoni & Jaroslaw Wroblewski | |

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

**Credited programs and projects**

Programs:

NewPGen by Paul Jobling.

APSieve by Michael Bell and Jim Fougeron.

tpsieve by Geoff Reynolds.

TwinGen by David Underbakke.

Srsieve by Geoffrey Reynolds.

PolySieve by
Robert Gerbicz.

PRP by George Woltman.

PrimeForm by the OpenPFGW group with George Woltman.

Proth.exe by Yves Gallot.

LLR by Jean Penné.

Primo (formerly Titanix) by Marcel Martin.

VFYPR by Tony Forbes.

FastECPP (unpublished program) by François Morain.

by Thomas Nguyen and other contributors.

The primality proving program is only credited above 300 digits.

Projects:

CP10 by Harvey Dubner, Tony Forbes, Nik Lygeros,
Michel Mizony and Paul Zimmermann.

GIMPS by George Woltman, Scott Kurowski, et al.

Twin Prime Search and PrimeGrid, coordinated by Michael
Kwok, Andrea Pacini, Rytis Slatkevicius.

Primecoin by Sunny King. The project does
sometimes not state discoverers and precise dates of records.

Riecoin

xN is an N-digit number which was used in a record but has no simple expression.

*x77* = 54538241683887582668189703590110659057865934764604873840781923513421103495579

67823802443758585361919599047776527963058419047009660578164772858363185263809

63039429596697403345980150176975651015797243607875744813349096932598456481621971387004081

245078011665141488718221144402728008841985546074620427235098669951080416452987007849969199215

172254155195529863097969853084190453728650037452220380184681391570092753684727879715473377537

*x397* = 3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473922496202015365392599420232189723690

2676229040360901005487309186655777663859063397693729163631275766077998753090384576371169385382793952602650644477477426123688904102021

71085974848375899782610469497787199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089

6778686512323214970638775969195989018082224873660480805597428344924204717278233388912711575086138672038905509854691940

8996533754855918734504345695472178776098869220527293364477300012589402397085151427849425715928696400187305151131481483

5359683991185108867465029353376453558170146445192639606953562379978691400614099121927887722702489794149145444471402905

843134098677487665910177188119952456912602688186634056637752516248275209509392841432650248795223050106564589182501

**Rules**

The listed number of digits is for the median prime, i.e. the i'th of 2i or 2i-1
primes. This prime determines the comparison to other sets.
Using the median is more fair but means the digit count may vary from pages for CC's and
BiTwins.

All primes must be above 10^{12} (mainly to avoid listing uninteresting k-tuplets
of tiny primes).

All primes must be proven, i.e. not just probable primes.

The record for k=1 is simply the largest known prime.

There are restrictions for k>1 to satisfy a reasonable interpretation of "simultaneous".

*Allowed* forms include:

Tuplets (primes as closely together as possible).

CC's (Cunningham Chains) of the 1st or 2nd kind.

BiTwins (chains of twin primes).

AP-k (k primes in arithmetic progression): Only allowed with difference k#
or p# or q#, where p and q are the smallest primes above k.

CPAP-k (k consecutive
primes in arithmetic progression): Same restriction as AP-k.

*Disallowed* forms include:

AP-k and CPAP-k with other differences than explicitly allowed above.

Generalized BiTwins (at bottom of linked page).

Generalized CC's (Cunningham Chains).

I am not aware of a previous comparison of this type and chose rules which seem
fair for the intended purpose. You can suggest rule changes, but arbitrary AP's will never be
allowed.

The rules for allowed forms were made with the idea that k numbers on a specific
predetermined form should be primes simultaneously. The form should usually only
be able to vary one thing, e.g. the first prime.

It should *not* be possible to compute a large set of primes and then find some subset
of k numbers with something in common, or choose a function which runs through k
already known primes.

An AP search usually first computes a large set of primes and then tests for AP's by
varying both one of the primes and the difference. This is much easier than
finding simultaneous primes. Extreme example: Any 2 numbers are in AP, so the 2
largest known primes will always form an AP-2 which is obviously disallowed (unless it has an allowed difference).

The longest known AP has 26 primes. The largest
known number of simultaneous primes satisfying the rules is an AP21 with
difference 19# and 20 digits, found in 2008. The smallest 18-tuplet above 100 has 25 digits and
may have been harder to find. The smallest AP's above 1000 are usually much smaller
than the smallest case of other forms, because there are more small candidates
without small prime factors. This makes it easier to find small examples of many
simultaneous primes for AP's. Sieving methods means this should not give AP's an
advantage when competing for the largest known case, when other forms have a
known case. A 21-tuplet with 29 digits was found in 2015. It may be the smallest
above 100.

**Submissions**

I would like to hear of all new records. Please mail any you find or know about,
but note the AP/CPAP rule. Say who should be credited, and which program proved
the primes if they are above 1000 digits.

If you are considering a form not mentioned then you can e-mail me and ask
whether it will be allowed. If you think you have found a clever form which
makes a search much easier than the explicitly allowed forms, then the form
almost certainly has a property that will not be allowed. Extreme example: The
n'th prime for k consecutive values of n.

Created and maintained by Jens Kruse Andersen,
jens.k.a@get2net.dk home