Last updated: 23 November 2024

Note: The original page was created by Jens Kruse Andersen (2005-2019).

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.

History Of Additions (2004 - today)

Introduction

Submissions

  • The largest known CPAP - 2 (only Cousin Primes & Sexy Prime Pairs) 

  • The largest known CPAP - k ; k = 3 .. 10

  • The largest known Sexy Prime Triples & Sexy Prime Quadruples

  • The minimal & the smallest known CPAP - k

  • The largest known CPAP - k difference (& History)

The largest known CPAP-k
k	Primes	n's	Digits	When	Discoverer(s)	History
3	17484430616589 • 254201 − 7 + 6 • n	0..2	16330	26 Jun 2024	Serge Batalov, EMsieve, OpenPFGW, CM
4	35734184537 • 11677# / 3 − 9 + 6 • n	0..3	5002	23 Jun 2024	Serge Batalov; EMsieve; Primo
5	2738129459017 • 4211# + 3399421517 + 30 • n	0..4	1805	31 Jan 2022	Serge Batalov
6	533098369554 • 2357# + 3399421517 + 30 • n	0..5	1012	13 Nov 2021	Serge Batalov
7	145706980166212 • 1069# + x253 + 210 • n	2..8	466	22 Nov 2021	Serge Batalov
8	8081110034864 • 619# + x253 + 210 • n	1..8	272	15 Nov 2021	Serge Batalov
9	7661619169627 • 379# + x153 + 210 • n	0..8	167	20 Nov 2021	Serge Batalov
10	189382061960492204 • 257# + x106 + 210 • n	0..9	121	9 Dec 2021	Serge Batalov

Big constants x106 = 1153762228327967262749742078637565852209646810567096822339169424875092523431859764709708315833909447378791 x153 = 9656383640115039654722740376098106958530576944745108587635040605371157826983 20398681243637298572057965220341992180981784112973206136355565433981118807417 x253 = 1161759929890532047130480253835658739849997983625515667103047375128118119991131225955 0734373874520536148519300924327947507674746679858816780182478724431966587843672408773 388445788142740274329621811879827349575247851843514012399313201211101277175684636727

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 + 6 • n 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 a CPAP-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.