Last updated: 20 July 2026

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, smallest known CPAP-k, and the largest known CPAP-k difference.

  •   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 - 2 (only Cousin Primes & Sexy Prime Pairs)  
  •   History Of Additions (2004 - today)  
  •   Introduction  
  •   Submissions  

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	2167681045402 • 4663# + 3399421517 + 30 • n	0..4	2006	10 Jun 2026	Serge Batalov; EMsieve; CM
6	533098369554 • 2357# + 3399421517 + 30 • n	0..5	1012	13 Nov 2021	Serge Batalov
7	216229301693757 • 1201# + x216a + 210 • n	0..6	518	21 Jun 206	Serge Batalov
8	329033474237114 • 761# + x215 + 210 • n	0..7	333	5 Jul 2026	Serge Batalov
9	5421105227721394 • 523# + x216b + 210 • n	1..9	223	20 Jul 2026	Serge Batalov
10	7059576465406145 • 311# + x124 + 210 • n	0..9	142	30 Jun 2026	Serge Batalov

Big constants x124 = 3299414364297973533431351613417061366766028861810156760941006093974566025024224093642242250594018968 464138302246979443638961 x215 = 4934529054201569602394992802571347201835957728850608808491674831929212579567713212950469099041808650 4160374890423905695940662712998474939478594072852653006483334863019764043599976605686734364772989908 539223019818949 x216a = 3170270091718350405893422628562865395898182006208709293502202456037281906234982949583411424372404277 6688948423367917328399415230717450311918278927562315603186953428020622792634052598132554643377947036 2962167933051651 x216b = 4312771350909941532324556748521165684704788405473442584536301010873113938974161392409803334205738947 3241150262930211996273857601376641975576306629205490705749810911280591525038888507020116110677840539 3832267631413701

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.