The Largest Known Simultaneous Primes - SP
Note: Original page was created by Jens Kruse Andersen (2005 - 2019).
This page is developed and maintained by Norman Luhn (since Oct 2021).
Contact: pzktupel[at]pzktupel[dot]de.
History Of Additions (2005 - today)
The largest known simultaneous primes
Big Constans
Introduction
Credited programs and projects
Rules
Submissions
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 + d, d = 0, 2, 6, 8 | 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 | 362079385668757696008683096558661746463 • 863# + 114023297140211 + d, d = 0, 2, 6, 8, 12, 18, 20, 26 |
Tuplet | 401 | 2023 | Michalis Christou, rieMiner | click |
9 | x93 • 541# + 145933845312371 + d, d = 0, 2, 6, 8, 12, 18, 20, 26, 30 | Tuplet | 312 | 2023 | Bielawski Mathematicians | click |
10 | x98 • 449# + 226554621544607 + d, d = 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 + d, d = 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 + d, d = 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 + d, d = 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 + d, d = 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 + d, d = 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 |
Big Constans
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 x93 = 182075127245948453356763852678412657384571384320476086323955359028566228121357180020362596219x98 = 14315614956030418747867488895208199566750873528908316976274174208238191434937011407287479676495550
x100 = 7620229574837377603687519001462575679324290703538353038451014140653366016375223441137843726785037341 x109 = 3662943827507055653453285926700023101620402654194921037456914703634367453333223968004841750810165461896894501 x155 = 573501051193549030504900787301835825160485101511983854806086469181338042238791
x397 = 3282186887886020104563334103168841560140170122832878265333984717524446848642006351778066196724473922496202015365392599420232189723690
2676229040360901005487309186655777663859063397693729163631275766077998753090384576371169385382793952602650644477477426123688904102021
71085974848375899782610469497787199182516499466558387976965904497393971453496036241885200541893611077817261813672809971503287259089
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.
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
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.