Note: This page was created by Jens Kruse Andersen (2004-2019).
Updated by Norman Luhn (since Oct 2021).
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)
The largest known simultaneous primes
Credited programs and projects
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.
|1||282589933-1||Mersenne||24862048||2018||Patrick Laroche, GIMPS|
|2||2996863034895 · 21290000 ± 1||Tuplet (twin)||388342||2016||Tom Greer, PrimeGrid, TwinGen, LLR|
|3||1128330746865 · 266439 · 2n - 1, n=0..2||CC, 1st kind||20013||2020||Dr. Michael Paridon||4||667674063382677 · 233608 - 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||23700 + 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# · 2n + 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# · 2n + 1, n=0..10||CC, 2nd kind||151||2016||Andrey Balyakin|
|12||906644189971753846618980352 · 233# · 2n + 1, n=0..11||CC, 2nd kind||123||2015||Andrey Balyakin|
|13||x84 · 61# · 2n - 1, n=0..12||CC, 1st kind||108||2014||Primecoin|
|14||x82 · 47# · 2n + 1, n=0..13||CC, 2nd kind||102||2014||Primecoin|
|15||14354792166345299956567113728 · 43# · 2n - 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 · 2n + 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
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.
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.
xN is an N-digit number which was used in a record but has no simple expression.
x77 = 54538241683887582668189703590110659057865934764604873840781923513421103495579x80 = 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
x397 = 3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473922496202015365392599420232189723690
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.
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,