The Largest Known Simultaneous Primes

Last updated: 06 March 2022

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.

The largest known simultaneous primes
k Primes Type Digits Year DiscovererRecord History
1 282589933-1 Mersenne 24862048 2018 Patrick Laroche, GIMPS
click
2 2996863034895 · 21290000 ± 1 Tuplet (twin) 388342 2016 Tom Greer, PrimeGrid, TwinGen, LLR
click
31128330746865 · 266439 · 2n - 1, n=0..2 CC, 1st kind 20013 2020 Dr. Michael Paridon
click
4 667674063382677 · 233608 - 1, +1, +5, +7 Tuplet 10132 2019 Peter Kaiser, PolySieve, LLR, Primo
click
5 585150568069684836 · 7757# / 85085 + 5 + d, d = 0, 2, 6, 8, 12 Tuplet 3344 2022 Peter Kaiser, OpenPFGW, Primo
click
6 23700 + 33888977692820810260792517451 + d, d = 0, 4, 6, 10, 12, 16 Tuplet 1114 2021 Vidar Nakling, Primo, Sixfinder
click
7 113225039190926127209 · 2339# / 57057 + 1 + d, d = 0, 2, 6, 8, 12, 18, 20 Tuplet 1002 2021 Peter Kaiser
click
8 2373007846680317952 · 761# · 2n + 1, n=0..7 CC, 2nd kind 338 2016 Andrey Balyakin
click
9 x100 · 503# + 220469307413891 + 0, 2, 6, 8, 12, 18, 20, 26, 30 Tuplet 309 2022 Bielawski Mathematicians
click
10 x98 · 449# + 226554621544607 + 0, 2, 6, 12, 14, 20, 24, 26, 30, 32 Tuplet 282 2021 Riecoin #1579367
click
11 341841671431409652891648 · 311# · 2n + 1, n=0..10 CC, 2nd kind 151 2016 Andrey Balyakin
click
12 906644189971753846618980352 · 233# · 2n + 1, n=0..11 CC, 2nd kind 123 2015 Andrey Balyakin
click
13 x84 · 61# · 2n - 1, n=0..12 CC, 1st kind 108 2014 Primecoin
click
14 x82 · 47# · 2n + 1, n=0..13 CC, 2nd kind 102 2014 Primecoin
click
15 14354792166345299956567113728 · 43# · 2n - 1, n=0..14 CC, 1st kind 47 2015 Andrey Balyakin
click
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
click
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
click
18 658189097608811942204322720 · 2n + 1, n=0..17 CC, 2nd kind 30 2014 Raanan Chermoni & Jaroslaw Wroblewski
click
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
click
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
click
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
click

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

x80 = 73853903764168979088206401473739410396455001112581722569026969860983656346568919

x81a = 566002435353389048470195154197633715327639809354150079355350346671860564824949963

x81b = 263663326886409378473341387047271336974122837948496277769621396327294641140893808

x81c = 223673331265817252994407640089592745163575915313761280958903819304727806835314518

x81d = 386727562407905441323542867468313504832835283009085268004408453725770596763660073

x82 = 5819411283298069803200936040662511327268486153212216998535044251830806354124236416

x83a = 39027761902802007714618528725397363585108921377235848032440823132447464787653697269

x83b = 10756750720700195380397697188448178460115725467111771468875842964723844354555016704

x83c = 61592551716229060392971860549140211602858978086524024531871935735163762961673908480

x84 = 106680560818292299253267832484567360951928953599522278361651385665522443588804123392

x89 = 44598464649019035883154084128331646888059795218766083584048621139159337786287845212160000

x98 = 14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550

x100 = 7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341

x109 = 3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501

x155 = 573501051193549030504900787301835825160485101511983854806086469181338042238791
67823802443758585361919599047776527963058419047009660578164772858363185263809

x177 = 2487800940974722434048969121852020452317389149426520567185777105403476571899339316926569
63039429596697403345980150176975651015797243607875744813349096932598456481621971387004081

x279 = 391920120903547650120329378696362044618319386407239508453167024439178985045498790600310298931
245078011665141488718221144402728008841985546074620427235098669951080416452987007849969199215
172254155195529863097969853084190453728650037452220380184681391570092753684727879715473377537

x397 = 3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473922496202015365392599420232189723690
2676229040360901005487309186655777663859063397693729163631275766077998753090384576371169385382793952602650644477477426123688904102021
71085974848375899782610469497787199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089

x586 = 4140119063956925627076259166420092791978438140163077033215492929339837973293152643271768110685094930596054056039071084
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 1012 (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