The Largest Known Simultaneous Primes

Note: This page was created by Jens Kruse Andersen (2004-2019).
Updated by Norman Luhn (since Oct 2021).
Contact: pzktupel[at]pzktupel[dot]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	Discoverer	Record 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
3	1128330746865 · 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

x397 = 3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473922496202015365392599420232189723690
2676229040360901005487309186655777663859063397693729163631275766077998753090384576371169385382793952602650644477477426123688904102021
71085974848375899782610469497787199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089

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