The Largest Known CPAP's

Last updated: 4 May 2022

Note: This page was created by Jens Kruse Andersen (2004-2019).
Updated by Norman Luhn (since Oct 2021).
Contact: pzktupel@pzktupel.de.

CPAP-k is short for k consecutive primes in arithmetic progression. For each value of k, this page maintains the largest known CPAP-k, the smallest known CPAP-k, and the largest known CPAP-k difference.

News
2022
May 4: Cousin Primes-record with 51934 digits by Serge Batalov
April 16: (sexy) CPAP-3 with 15004 digits by Serge Batalov
January 31: CPAP-5-record with 1805 digits by Serge Batalov

History Of Additions (2004-2021)

Introduction
Submissions

The largest known CPAP-2 (Twin Primes, Cousin Primes, Sexy Prime Pairs, ... )

The largest known CPAP-k for k=3..10

The largest known sexy CPAP's

The minimal & the smallest known CPAP-k

The largest known CPAP-k difference (& History)

Big constants

Credited programs
Other pages

Introduction
A prime number is a natural number which only has the two divisors 1 and itself. The first are 2, 3, 5, 7, 11.
An AP-k is any case of k primes in arithmetic progression, i.e. of the form p+d·n for some d (the difference between the primes) and k consecutive values of n.
Example: 41 + 6n for n = 0, 1, 2, 3 gives the AP-4 41, 47, 53, 59. See Primes in Arithmetic Progression Records for the largest and smallest AP-k.

A CPAP-k is an AP-k where the k primes are consecutive, i.e. there are no other primes between them. (CPAP can mean many other things outsidemathematics).
The AP-4 41, 47, 53, 59 is not a CPAP-4 because 43 is also prime. But 47, 53, 59 is aCPAP-3. This page is only about CPAP-k.

A CPAP-k search often has two parts: Find an AP-k and then test whether the k primes are consecutive. If the difference between the primes is small then it is sometimes possible to make sure in advance that all intermediate numbers will be composite.

k# (called k primorial) is the product of all primes ≤ k, e.g. 10# = 2 · 3 · 5 · 7 = 210.2# = 2, 3# = 6, 5# = 30, 7# = 210, 11# = 2310
The prime difference in an AP-k (and thus a CPAP-k) must be a multiple of k# to avoid factors ≤ k, assuming the primes in the AP-k are above k.
Avoiding intermediate primes in a CPAP-k becomes harder when the prime difference is big, so many searches only try for difference k#.
A CPAP-6 has minimal difference 6# = 30 which is low in this context. CPAP-7 to -10 all have minimal difference 10# = 7# = 210 which makes it harder.
However it is possible to make a guarantee against intermediate primes in a CPAP-7 larger than around 190 digits. x177 has been used for this.

A CPAP-11 would have minimal difference 11# = 2310. This seems extremely hard to find and nobody has even tried as far as I know.
With current methods it may take trillions of cpu GHz years according to the people who found the first known CPAP-10.

It seems likely that there are infinitely many CPAP-k with prime difference c·k#, for all c and k. You will be famous (among mathematicians anyway) by proving this, because the proof would probably cover lots of other cases, e.g. the k-tuple conjecture. k=2 and c=1 gives the twin prime conjecture, enough for fame.
Ben Green & Terence Tao presented a proof in 2004 that The primes contain arbitrarily long arithmetic progressions, but their result is not about consecutive primes.

Submissions
I would like to hear of all CPAP's which make one of the record tables. Please mail any you find or know about. Say who should get credit and how the primes were proved. The tables are not for numbers which are only prp's (probable primes). I have software to prove prp's up to a few thousand digits. You can submit CPAP's consisting of prp's but I want shared credit for performing proofs above 2000 digits.
If a CPAP was found with an expression involving a small or big constant then please give the expression and constant, not just a decimal expansion of the primes.

A link on the year of a record is to an announcement of that record.
A link on "Primo" was to Primo certificates of primality until 23 January 2009 where the website moved. Most of the certificates are currently offline. They are available by email request.

The largest known CPAP-k for each k
k Primes n's Digits Year Discoverer(s)
Record History
3
2494779036241 · 249800 + 1 + 6n n=0..2 15004 2022 Serge Batalov
click
4
62399583639 · 9923# - 3399421607 + 30n n=0..3 4285 2021 Serge Batalov
click
5
2738129459017 · 4211# + 3399421517 + 30n n=0..4 1805 2022 Serge Batalov
click
6
533098369554 · 2357# + 3399421517 + 30n n=0..5 1012
2021 Serge Batalov
click
7
145706980166212 · 1069# + x253 + 210n n=2..8 466 2021 Serge Batalov
click
8
8081110034864 · 619# + x253 + 210n n=1..8 272 2021 Serge Batalov
click
9
7661619169627 · 379# + x153 + 210n n=0..8 167 2021 Serge Batalov
click
10
189382061960492204 · 257# + x106 + 210n n=0..9

121

2021 Serge Batalov
click

Big constants

Some CPAP searches uses big numbers with no simple expression, chosen to guarantee many composites.
xN is an N-digit constant used in the above CPAP's:

x25 = 4951280491824700272223109

x32a = 19111438098711663697781258214361

x32b = 23889297537258134291826489698341

x34 = 7237338580293614937416191022926747

x60 = 185753346262303075693548200157509979962582683099948651188109

x65 = 87103490338886343449123705322656962705040008760706629856986802283

x72 = 976380866068288254300174899353741982908376159786126663199157748266010387

x76 = 6240141611007307622465889025426185177074468140120944390087327315890659848721

x77 = 54538241683887582668189703590110659057865934764604873840781923513421103495579

x87 = 279872509634587186332039135414046330728180994209092523040703520843811319320930380677867

x92 = 43804034644029893325717710709965599930101479007432825862362446333961919524977985103251510661

x97 = 1089533431247059310875780378922957732908036492993138195385213105561742150447308967213141717486151

x99 = 158794709618074229409987416174386945728371523590452459863667791687440944143462160821328735143564091

x106 = 1153762228327967262749742078637565852209646810567096822339169424875092523431859764709708315833909447378791

x133 = 220505805098423836819764228021381624235600014226631323732190862768
0719158137585667750659950011921258708497882049342113679838524318399

x153 = 9656383640115039654722740376098106958530576944745108587635040605371157826983
20398681243637298572057965220341992180981784112973206136355565433981118807417

x155 = 57350105119354903050490078730183582516048510151198385480608646918133804223879
167823802443758585361919599047776527963058419047009660578164772858363185263809

x156 = 151188159161413910473224635599091717347191347475741199123999487827863323498969
128759882372032969740473050518757788386432708639912852688724160336908656571679

x177 = 2487800940974722434048969121852020452317389149426520567185777105403476571899339316926569
63039429596697403345980150176975651015797243607875744813349096932598456481621971387004081

x218 = 1710314864346465484108592184076855564975234702180394797683443122161589881975347244280715286678546006913815669
5534325982537058769170527990893016488272955332936078978017144057593234210769778622909436276234673417088036739

x253 = 1161759929890532047130480253835658739849997983625515667103047375128118119991131225955
0734373874520536148519300924327947507674746679858816780182478724431966587843672408773
388445788142740274329621811879827349575247851843514012399313201211101277175684636727

x272 = 5664766322442249254371230091507914625954793773584708235455123610906765158309227090915170593
0326496339047403010122690110808765199726179853671275222228963017234966647316007126105443546
537713130920430017652260314022049007616504227665344802771835420837602700201417124972030829

x371 = 211456793391168128047279038344199047999237962856056679635442435783875508364793549383096908674
055907232062402104302800309850677157870519355607544687390457996561399556465527843958680778163
142462848746567536695606514797322232795128407336312883815163380914875924413574595458906498410
64655307278791070721577180306286411790729833109653930842047229740770849707899029657847082299

x498 = 1161719015143012794121754001566538080263593537397930820975927707160538253075959973861229736824601862
1756915012917466523258116732226677456532069983002381169824910734505346698565401855618416880928412560
3664216792484911826057321448712777916321546744979037962370826360519799915114703062304116721936655626
1929499932561340830714609010093682606701953247557023517250162447697065051646724698400705091689802253
52420335236903442680040421006151255566837889765262056304737646834967091494264642282692101351482213

x632 = 2945339776545027154539918808526655368252378620585099496385650600431498269661083903162110429127353107760157
5722896273706142925617722759452435429488328389328281466289664367352954006161792070955769212597757921750265
7961793687809965941432837668975308693630297479962123616982055909919099170254969337754185770956958979321362
7618473506498212549583475520940170609152997656163627242028405951204832924677679233522715633270907575099531
8190876684457108535835673100713235902439791043089273743933820748006776935061385300428903923277275802622905
803642678199081441811796580012014897740453191957526038333320588240996195703518136355252551601080488639

x2506 = 5082567793.....2515921377


Credited programs
The primality proving program is only credited above 300 digits.
CP09 was a program/project by Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony and Paul Zimmermann.
CP10 was by the same people as CP09.
Primo (formerly Titanix) by Marcel Martin.
VFYPR by Tony Forbes.
PrimeForm by the OpenPFGW group with George Woltman.
NewPGen by Paul Jobling.
APSieve by Michael Bell and Jim Fougeron.
Proth.exe by Yves Gallot.
FastECPP by François Morain, Jens Franke, Thorsten Kleinjung and Tobias Wirth.
TwinGen by David Underbakke.
LLR by Jean Penné.
APTreeSieve by Jens Kruse Andersen.

Other pages
CPAP's in The Prime Pages
Programs to search and prove large primes
Project to find the first known CPAP-10
Official announcement of CPAP-10 discovery
Manfred Toplic's page
Primes in arithmetic progression in Wikipedia
The Largest Known CPAP-3

Pages with similar records
Primes in Arithmetic Progression Records
Prime k-tuplets
Cunningham Chain records
BiTwin records
The Largest Known Simultaneous Primes

Sources
E-mail correspondence with some discoverers.
Announcements linked in the tables.
Titanic CPAP's: http://primes.utm.edu/top20/page.php?id=13.

This page is at http://primerecords.dk/cpap.htm and licensed under the GFDL.
Created and maintained by Jens Kruse Andersen, jens.k.a@get2net.dk   home