The Largest Known CPAP's

CPAP-k is short for k consecutive primes in arithmetic progression. For each value of k, this page maintains the 10 largest known CPAP-k, the smallest known CPAP-k, and the largest known CPAP-k difference.
News
Introduction
Submissions
The largest known CPAP-k for each k
The largest known CPAP-3
The largest known CPAP-4
The largest known CPAP-5
The largest known CPAP-6
The largest known CPAP-7
The largest known CPAP-8
The largest known CPAP-9
The largest known CPAP-10
The largest known sexy CPAP's
The minimal CPAP-k
The smallest known CPAP-k
The largest known CPAP-k difference
History of the largest known CPAP-k difference
CPAP-2
Big constants
Credited programs
Other pages

News (only records)
2019
October 11: New sexy and general CPAP-4 record with 3025 digits by Norman Luhn & Gerd Lamprecht, PrimeForm, Primo.
October 7: New sexy CPAP-3 record with 6701 digits by Norman Luhn & Gerd Lamprecht, APSieve, Primo.
October 4: New sexy CPAP-4 record with 2053 digits by Ken Davis, APTreeSieve, PrimeForm, Primo.
October 1: New sexy CPAP-3 record with 6180 digits by Ken Davis, APTreeSieve, PrimeForm.
September 11: New sexy CPAP-4 record with 2001 digits by Gerd Lamprecht, Primo.
August 13: New sexy CPAP-3 record with 6116 digits by Gerd Lamprecht, APSieve, PrimeForm, Primo.
June 24: New sexy CPAP-4 record with 1534 digits by Peter Kaiser.
May 13: New sexy CPAP-3 record with 6031 digits by Peter Kaiser.
May 11: New sexy CPAP-4 record with 1138 digits by Marek Hubal, PrimeForm, Primo.

2018
November 11: Smallest known CPAP-7 with 23 digits by Paul Zimmermann.
September 17: New sexy CPAP-3 record with 5320 digits by Gerd Lamprecht, APSieve, PrimeForm, Primo.
September 15: Smallest known CPAP-7 with 24 digits by Paul Zimmermann.
September 7: New sexy CPAP-4 record with 1034 digits by Gerd Lamprecht, APSieve, PrimeForm, Primo.
August 24: A second smallest known CPAP-7 with 25 digits by Paul Zimmermann.
August 3: Smallest known CPAP-7 with 26, again 26 and then 25 digits by Paul Zimmermann.
July 27: Smallest known CPAP-7 with 27 and again 27 digits by Paul Zimmermann.
July 20: Smallest known CPAP-7 with 30, 30, 28, 28 and finally 27 digits by Paul Zimmermann.
July 18: New CPAP-5 record with 1290 digits by Norman Luhn, Gerd Lamprecht, APSieve, PrimeForm, Primo.
July 17: New CPAP-4 record with 3024 digits by Gerd Lamprecht, APSieve, PrimeForm, Primo.
July 13: Smallest known CPAP-7 with 30 digits by Paul Zimmermann.
July 6: A second smallest known CPAP-7 with 31 digits by Paul Zimmermann.
July 5: Smallest known CPAP-7 with 31 digits by Paul Zimmermann.
July 4: A third smallest known CPAP-7 with 32 digits by Paul Zimmermann.
July 1: A second smallest known CPAP-7 with 32 digits by Paul Zimmermann.
June 27: Smallest known CPAP-7 with 32 digits by Paul Zimmermann.
May 8: New CPAP-6 record with 427 digits by Gerd Lamprecht, APSieve, PrimeForm, Primo.

2014
March 19: A second CPAP-3 record with 10546 digits by David Broadhurst, PrimeForm, Primo.
February 21: New CPAP-3 record with 10546 digits by David Broadhurst, PrimeForm, Primo.
February 1: A third CPAP-3 record with 10545 digits by David Broadhurst, PrimeForm, Primo.
January 11: A second CPAP-3 record with 10545 digits by David Broadhurst, PrimeForm, Primo.

2013
December 17: New CPAP-3 record with 10545 digits by David Broadhurst, PrimeForm, Primo.
November 5: New CPAP-3 record with 10042 digits by Jens Kruse Andersen, Pierre Cami, Ken Davis, NewPGen, PrimeForm, Primo.
November 1: New CPAP-5 record with 1209 digits by David Broadhurst, PrimeForm, Primo.
October 31: A third CPAP-3 record with 7535 digits by David Broadhurst, Bouk de Water, PrimeForm, Primo.
October 27: A second CPAP-3 record with 7535 digits by David Broadhurst, Bouk de Water, PrimeForm, Primo.
October 25: A second CPAP-4 record with 3021 digits by David Broadhurst, PrimeForm, Primo.
October 25: New CPAP-4 record with 3021 digits by David Broadhurst, PrimeForm, Primo.
October 23: A second CPAP-4 record with 3020 digits by David Broadhurst, PrimeForm, Primo.
October 21: New CPAP-4 record with 3020 digits by David Broadhurst, PrimeForm, Primo.
October 21: New CPAP-3 record with 7535 digits by David Broadhurst, Bouk de Water, PrimeForm, Primo.
October 16: A second CPAP-4 record with 2575 digits by David Broadhurst, PrimeForm, Primo.
October 13: New CPAP-4 record with 2575 digits by David Broadhurst, PrimeForm, Primo.
October 12: A second CPAP-4 record with 2574 digits by David Broadhurst, PrimeForm, Primo.
October 10: New CPAP-4 record with 2574 digits by David Broadhurst, PrimeForm, Primo.

2012
December 10: New CPAP-4 record with 2148 digits by Jim Fougeron, Primo.

2008
December 8: A second CPAP-10 record with 93 digits by Manfred Toplic, CP10. 10 years after the first.

2007
December 26: New CPAP-7 record with 266 digits by Jens Kruse Andersen.
November 12: New CPAP-4 record with 1777 digits and record difference 2880 by Ken Davis, NewPGen, Primo.
October 1: CPAP-3 record for probable primes allowed with 10042 digits by Jens Kruse Andersen, Ken Davis, NewPGen, PrimeForm.
January 7: New sections: The largest known CPAP-k for each k and CPAP-2.

2006
No new records.

2005
March 19: New CPAP-9 record with 101 digits (same as before) by Hans Rosenthal & Jens Kruse Andersen.
February 27: New CPAP-3 record with 7535 digits by David Broadhurst, François Morain, FastECPP, PrimeForm.

2004
December 31: New CPAP-5 with record difference 2310 by Jim Fougeron, Primo.
December 3: New CPAP-4 with record difference 2310 by Jens Kruse Andersen, PrimeForm, Primo.
December 1: New CPAP-4 with record difference 2004 by Jim Fougeron.
November 30: New record table with The largest known CPAP-k difference, containing new records for CPAP-3, -5, -6 and -7 by Torbjörn Alm & Jens Kruse Andersen.
November 18: New CPAP-3 record with 7402 digits.
November 18: This news section is started (this page opened September 5 2003).

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)
3 1213266377 · 235000 - 1 + 2430n n=0..2 10546 2014 David Broadhurst, PrimeForm, Primo
4 121152729080 · 7019#/1729 + 1 + 6n n=0..3 3025 2019 Gerd Lamprecht & Norman Luhn, PrimeForm, Primo
5 2746496109133 · 3001# + 26891 + 30n n=0..4 1290 2018 Norman Luhn, Gerd Lamprecht, APSieve, PrimeForm, Primo
6 386140564676 · 1000# + 26861 + 30n n=0..5 427
 2018 Gerd Lamprecht, APSieve, PrimeForm, Primo
7 4785544287883 · 613# + x253 + 210n n=0..6 266 2007 Jens Kruse Andersen
8 10097274767216 · 250# + x99 + 210n n=0..7 112 2003 Jens Kruse Andersen
9 73577019188277 · 199#·227·229 + x87 + 210n n=0..8 101 2005 Hans Rosenthal & Jens Kruse Andersen
10 1180477472752474 · 193# + x77 + 210n n=0..9

93

 2008 Manfred Toplic, CP10

The largest known CPAP-3
Rank Primes n's Digits Year Discoverer(s)
1 1213266377 · 235000 - 1 + 2430n n=0..2 10546 2014 David Broadhurst, PrimeForm, Primo
2 1043085905 · 235000 - 18199 + 18198n n=0..2 10546 2014 David Broadhurst, PrimeForm, Primo
3 109061779 · 235003 - 1 + 5928n n=0..2 10545 2014 David Broadhurst, PrimeForm, Primo
4 350049825 · 235000 - 1 + 3852n n=0..2 10545 2014 David Broadhurst, PrimeForm, Primo
5 146462479 · 235001 - 8767 + 8766n n=0..2 10545 2013 David Broadhurst, PrimeForm, Primo
6 19067408 · 233333 - 12539 + 6270n n=0..2 10042 2013 Jens Kruse Andersen, Pierre Cami, Ken Davis, NewPGen, PrimeForm, Primo
7 90568041 · 225077 - 9001 + 4500n n=0..2 7557 2013 Pierre Cami, PrimeForm, Primo
8 49787793 · 225003 - 1 + 11634n n=0..2 7535 2013 David Broadhurst, Bouk de Water, PrimeForm, Primo
9 368523657 · 225000 - 14161 + 7080n n=0..2 7535 2013 David Broadhurst, Bouk de Water, PrimeForm, Primo
10 122248047 · 225001 - 15661 + 7830n n=0..2 7535 2013 David Broadhurst, Bouk de Water, PrimeForm, Primo

The largest known CPAP-4
Rank Primes n's Digits Year Discoverer(s)
1 121152729080 · 7019#/1729 + 1 + 6n n=0..3 3025 2019 Gerd Lamprecht & Norman Luhn, PrimeForm, Primo
2 55072065656 · 7013# + 9843049 + 30n n=0..3 3024 2018 Gerd Lamprecht, APSieve, PrimeForm, Primo
3
 62037039993 · 7001# + 7811555723 + 30n n=0..3 3021 2013 David Broadhurst, PrimeForm, Primo
4
 50946848056 · 7001# + 7811555723 + 30n n=0..3 3021 2013 David Broadhurst, PrimeForm, Primo
5
 26997933312 · 7001# + 7811555663 + 30n n=0..3 3020 2013 David Broadhurst, PrimeForm, Primo
6
 25506692100 · 7001# + 7811555693 + 30n n=0..3 3020 2013 David Broadhurst, PrimeForm, Primo
7
 198267970563 · 6007# + 7811555663 + 30n n=0..3 2575 2013 David Broadhurst, PrimeForm, Primo
8
 153104252515 · 6007# + 7811555663 + 30n n=0..3 2575 2013 David Broadhurst, PrimeForm, Primo
9
 60213811936 · 6007# + 7811555693 + 30n n=0..3 2574 2013 David Broadhurst, PrimeForm, Primo
10
 24576954772 · 6007# + 7811555723 + 30n n=0..3 2574 2013 David Broadhurst, PrimeForm, Primo

The largest known CPAP-5
Rank Primes n's Digits Year Discoverer(s)
1 2746496109133 · 3001# + 26891 + 30n n=0..4 1290 2018 Norman Luhn, Gerd Lamprecht, APSieve, PrimeForm, Primo
2
 406463527990 · 2801# + 1633050283 + 30n n=0..4 1209 2013 David Broadhurst, PrimeForm, Primo
3
 993530619517 · 2503# + 1633050253 + 30n n=0..4 1073 2013 David Broadhurst, PrimeForm, Primo
4
 495690450643 · 2503# + 1633050283 + 30n n=0..4 1072 2013 David Broadhurst, PrimeForm, Primo
5
 150822742857 · 2503# + 1633050253 + 30n n=0..4 1072 2013 David Broadhurst, PrimeForm, Primo
6
 94807777362 · 2503# + 1633050253 + 30n n=0..4 1072 2013 David Broadhurst, PrimeForm, Primo
7
 142661157626 · 2411# + 71427757 + 30n n=0..4 1038 2002 Jim Fougeron, Primo
8
 9400734826 · 1499# + x632 + 2310n n=0..4 645 2004 Jim Fougeron, Primo
9
 60677170100 · 859# + 1204600427 + 30n n=0..4 370 2003 Jens Kruse Andersen, Jim Fougeron, Primo
10
 51892636811 · 859# + 1204600427 + 30n n=0..4 370 2003 Jens Kruse Andersen, Jim Fougeron, Primo

The largest known CPAP-6
Rank Primes n's Digits Year Discoverer(s)
1 386140564676 · 1000# + 26861 + 30n n=0..5 427
 2018 Gerd Lamprecht, APSieve, PrimeForm, Primo
2
 44770344615 · 859# + 1204600427 + 30n n=0..5 370 2003 Jens Kruse Andersen, Jim Fougeron, Primo
3
 2121022995 · 859# + 1204600427 + 30n n=0..5 369 2003 Jim Fougeron, VFYPR
4
 200605912613 · 700# + 2437704499910036551539466113407 + 30n n=0..5 301 2003 Torbjörn Alm & Jens Kruse Andersen, VFYPR
5
 4785544287883 · 613# + x253 + 210n n=1..6 266 2007 Jens Kruse Andersen
6
 4785544287883 · 613# + x253 + 210n n=0..5 266 2007 Jens Kruse Andersen
7
 3479345458420 · 613# + x253 + 210n n=0..5 266 2007 Jens Kruse Andersen
8
 3286247932804 · 613# + x253 + 210n n=0..5 266 2007 Jens Kruse Andersen
9
 3149164299724 · 613# + x253 + 210n n=0..5 266 2007 Jens Kruse Andersen
10
 3055250619386 · 613# + x253 + 210n n=0..5 266 2007 Jens Kruse Andersen

The largest known CPAP-7
Rank Primes n's Digits Year Discoverer(s)
1 4785544287883 · 613# + x253 + 210n n=0..6 266 2007 Jens Kruse Andersen
2 194688251849 · 601# + x155 + 210n n=0..6 259 2003 Jim Fougeron
3 7127829789350 · 460# + x177 + 210n n=0..6 201 2003 Torbjörn Alm & Jens Kruse Andersen
4 7091037255023 · 460# + x177 + 210n n=0..6 201 2003 Torbjörn Alm & Jens Kruse Andersen
5 7032468079552 · 460# + x177 + 210n n=0..6 201 2003 Torbjörn Alm & Jens Kruse Andersen
6 10097274767216 · 250# + x99 + 210n n=1..7 112 2003 Jens Kruse Andersen
7 10097274767216 · 250# + x99 + 210n n=0..6 112 2003 Jens Kruse Andersen
8 10080119713620 · 250# + x99 + 210n n=0..6 112 2003 Jens Kruse Andersen
9 10074509638938 · 250# + x99 + 210n n=0..6 112 2003 Jens Kruse Andersen
10 10003305227518 · 250# + x99 + 210n n=0..6 112 2003 Jens Kruse Andersen

The largest known CPAP-8
Rank Primes n's Digits Year Discoverer(s)
1 10097274767216 · 250# + x99 + 210n n=0..7 112 2003 Jens Kruse Andersen
2 121333717075627 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen
3 120293524089244 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen
4 119845733530287 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen
5 117686134150353 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen
6 109571139518701 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen
7 104072945625210 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen
8 101107729483324 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen
9 73577019188277 · 199#·227·229 + x87 + 210n n=1..8 101 2005 Hans Rosenthal & Jens Kruse Andersen
10 73577019188277 · 199#·227·229 + x87 + 210n n=0..7 101 2005 Hans Rosenthal & Jens Kruse Andersen

The largest known CPAP-9
Rank Primes n's Digits Year Discoverer(s)
1 73577019188277 · 199#·227·229 + x87 + 210n n=0..8 101 2005 Hans Rosenthal & Jens Kruse Andersen
2 25127352416973 · 199#·227·229 + x87 + 210n n=0..8 101 2004 Hans Rosenthal & Jens Kruse Andersen
3 1180477472752474 · 193# + x77 + 210n n=1..9

93

 2008 Manfred Toplic, CP10
4 1180477472752474 · 193# + x77 + 210n n=0..8

93

 2008 Manfred Toplic, CP10
5 1178322910173399 · 193# + x77 + 210n n=1..9

93

 2008 Manfred Toplic, CP10
6 1099212552579171 · 193# + x77 + 210n n=1..9

93

 2007 Manfred Toplic, CP10
7 1056761470535795 · 193# + x77 + 210n n=0..8

93

 2006 Manfred Toplic, CP10
8 1028784711070427 · 193# + x77 + 210n n=1..9

93

 2006 Manfred Toplic, CP10
9 580596232159174 · 193# + x77 + 210n n=0..8 93 1998 Robert Piche, CP10
10 507618446770482 · 193# + x77 + 210n n=1..9 93 1998 Manfred Toplic, CP10

The largest (and only) known CPAP-10
Rank Primes n's Digits Year Discoverer(s)
1 1180477472752474 · 193# + x77 + 210n n=0..9

93

 2008 Manfred Toplic, CP10
2 507618446770482 · 193# + x77 + 210n n=0..9

93

 1998 Manfred Toplic, CP10

Sexy primes are two primes separated by 6. This can be extended to sexy triplets and sexy quadruplets, but not further due to divisibility by 5 - apart from the single non-consecutive quintuplet (5, 11, 17, 23, 29). The definition does not require consecutive primes but the records have it. The below table shows the record history since 2010.
The largest known sexy CPAP's
k Primes n's Digits Year Discoverer(s)
3
22582235875 · 222224 + 1 + 6n n=0..2 6701 2019 Norman Luhn & Gerd Lamprecht, APSieve, Primo
3 (72865897 · 809857 · 4801# · (809857 · 4801# + 1) + 210) · (809857 · 4801# - 1) / 35 + 1 + 6n n=0..2 6180 2019 Ken Davis, PrimeForm, APTreeSieve
3
20730011943 · 14221# + 344231 + 6n n=0..2 6116 2019 Gerd Lamprecht, APSieve, PrimeForm, Primo
3
10409207693 · 220000 - 1 + 6n n=0..2 6031
 2019 Peter Kaiser
3
12909901425 · 12391# + 344237 + 6n n=0..2 5320 2018 Gerd Lamprecht, APSieve, PrimeForm, Primo
3 (84055657369 · 205881 · 4001# · (205881 · 4001# + 1) + 210) · (205881 · 4001# - 1) / 35 + 1 + 6n n=0..2 5132 2006 Ken Davis, PrimeForm, APTreeSieve

4 121152729080 · 7019#/1729 + 1 + 6n n=0..3 3025 2019 Gerd Lamprecht & Norman Luhn, PrimeForm, Primo
4 1901870849 · (269504 · 1601# · (269504 · 1601# + 1) · (269504 · 1601# - 1) / 385) + 6 · (269504 · 1601# - 1) - 5 + 6n n=0..3 2053 2019 Ken Davis, APTreeSieve, PrimeForm, Primo
4 46329193156 · 4657# + 344231 + 6n n=0..3 2001 2019 Gerd Lamprecht, Primo
4 19299420002127 · 25050 + 17233 + 6n n=0..3 1534 2019 Peter Kaiser, PrimeForm, Primo
4 1567237911 · 2677# + 3301 + 6n n=0..3 1138 2019 Marek Hubal, PrimeForm, Primo
4 37675946750 · 2400# + 344231 + 6n n=0..3 1034 2018 Gerd Lamprecht, APSieve, PrimeForm, Primo
4 23333 + 1582534968299 + 6n n=0..3 1004 2010 Ken Davis, PrimeForm, Primo

The minimal CPAP-k
k Primes n's Digits Year Discoverer(s)
3 3 + 2n n=0..2 1    
4 251 + 6n n=0..3 3    
5 9843019 + 30n n=0..4 7    
6 121174811 + 30n n=0..5 9 1967 L. J. Lander & T. R. Parkin

The minimal CPAP-k is currently only known for k<7. After that the prime difference must be at least 210 and the minimal solution is probably so large that an exhaustive search for it would be extremely hard. Heuristics (estimates based on probability) indicate the minimal CPAP-7 may have around 21 digits. As of November 2018, the smallest known has 23 digits.

The need for at least 209 · 6 = 1254 composites in a CPAP-k with k>6 means it is much harder to find CPAP's with small primes than larger ones. The below table shows the 3 smallest known CPAP-k when the minimal is unknown. There are only two known CPAP-10.
The smallest known CPAP-k
k Primes n's Digits Year Discoverer(s)
7(1) 71137654873189893604531 + 210n n=0..6 23
 2018 Paul Zimmermann
7(2)382003672700092872707633 + 210nn=0..624
2018Paul Zimmermann
7(3) 2210835776623037377907953 + 210n n=0..6 25
 2018 Paul Zimmermann
8(1) 2799806429564 · 83#·113 + x34 + 210n n=0..7 47 2004 Hans Rosenthal & Jens Kruse Andersen
8(2) 5351738881202 · 83#·113 + x34 + 210n n=0..7 48 2004 Hans Rosenthal & Jens Kruse Andersen
8(3) 16003606986539 · 83#·113 + x34 + 210n n=0..7 48 2004 Hans Rosenthal & Jens Kruse Andersen
9(1) 3416716311814 · 179#/(149·157) + x65 + 210n n=0..8 79 2004 Hans Rosenthal & Jens Kruse Andersen
9(2) 12606057030290 · 179#/(149·157) + x65 + 210n n=0..8 80 2005 Hans Rosenthal & Jens Kruse Andersen
9(3) 52515434335080 · 179#/(149·157) + x65 + 210n n=0..8 80 2005 Hans Rosenthal & Jens Kruse Andersen
10(1) 507618446770482 · 193# + x77 + 210n n=0..9

93

 1998 Manfred Toplic, CP10
10(2) 1180477472752474 · 193# + x77 + 210n n=0..9

93

 2008 Manfred Toplic, CP10

Finding a CPAP-k with large difference is harder because more numbers must be composite at the same time. The table shows the single largest known CPAP-k difference (in bold) for each k. If more than one CPAP is known with that difference then only the first found is shown. Only the minimal difference k# = 210 has been found for CPAP-8, -9 and -10.
The largest known CPAP-k difference
k Primes n's Digits Year Discoverer(s)
3 x2506 + 21102n n=0..2 2506 2004 Torbjörn Alm & Jens Kruse Andersen, PrimeForm, Primo
4 25900 + 469721931951 + 2880n n=0..3 1777 2007 Ken Davis, NewPGen, PrimeForm, Primo
5 9400734826 · 1499# + x632 + 2310n n=0..4 645 2004 Jim Fougeron, Primo
6 x218 + 840n n=0..5 218 2004 Torbjörn Alm & Jens Kruse Andersen
7 x133 + 420n n=0..6 133 2004 Torbjörn Alm & Jens Kruse Andersen
8 x92 + 210n n=0..7 92 1997 Harvey Dubner, Tony Forbes & Paul Zimmermann
9 500996388736659 · 193# + x76 + 210n n=0..8 92 1998 Manfred Toplic, CP09
10 507618446770482 · 193# + x77 + 210n n=0..9

93

 1998 Manfred Toplic, CP10

The following table shows current and old records, some from before the record category was added here. The first discovered CPAP-7, -8, -9 and -10 are all listed. I would guess the first CPAP-7 was also the first CPAP-5 and -6 with difference 210, and that no larger difference was known until the listed records.
Hans Rosenthal's A probable CPAP-3 with d=17676 is not listed because no primality proofs were performed.
History of the largest known CPAP-k difference
k Primes n's Digits Year Discoverer(s)
3 x2506 + 21102n n=0..2 2506 2004 Torbjörn Alm & Jens Kruse Andersen, PrimeForm, Primo
3 41#137 - 5576107 + 10164n n=0..2

1985

 2002 David Broadhurst, Primo
 
4 25900 + 469721931951 + 2880n n=0..3 1777 2007 Ken Davis, NewPGen, PrimeForm, Primo
4 46313478 · 1201#/1302643 + x498 + 2310n n=0..3 505 2004 Jens Kruse Andersen, PrimeForm, Primo
4 78006074 · 883# + x371 + 2004n n=0..3 379 2004 Jim Fougeron
4 23320 + 1308319536235 + 1470n n=0..3 1000 2004 Hans Rosenthal, Jim Fougeron, Primo
 
5 9400734826 · 1499# + x632 + 2310n n=0..4 645 2004 Jim Fougeron, Primo
5 x272 + 1350n n=0..4 272 2004 Jens Kruse Andersen
 
6 x218 + 840n n=0..5 218 2004 Torbjörn Alm & Jens Kruse Andersen
 
7 x133 + 420n n=0..6 133 2004 Torbjörn Alm & Jens Kruse Andersen
7 x97 + 210n n=0..6 97 1995 Harvey Dubner & Harry Nelson
 
8 x92 + 210n n=0..7 92 1997 Harvey Dubner, Tony Forbes & Paul Zimmermann
 
9 500996388736659 · 193# + x76 + 210n n=0..8 92 1998 Manfred Toplic, CP09
 
10 507618446770482 · 193# + x77 + 210n n=0..9

93

 1998 Manfred Toplic, CP10

CPAP-2
CPAP-k usually assumes k > 2, but otherwise a CPAP-2 is by definition any pair of consecutive primes. Other parts of this page do not include CPAP-2. There are infinitely many primes and thus infinitely many CPAP-2.
Only proved primes are allowed on this page, so the top-10 CPAP-2 are the top-10 cases of 2 consecutive proved primes. In 2013 (and probably for decades before that) this is the top-10 twin primes, due to limitations in searched prime forms and known primality proving methods. The largest is 2996863034895  ·  21290000 ± 1 with 388342 digits, found in 2016 by Tom Greer, PrimeGrid, TwinGen, LLR.
1 is not considered a prime, so the minimal CPAP-2 is 2, 3. This is an exception to the rule that a CPAP-k difference must be a multiple of k#. The exception is only possible because this CPAP-k starts with k. The only other exception is the CPAP-3 with primes 3, 5, 7.
The largest known CPAP-2 difference is the largest known prime gap between 2 proved primes. This is a gap of 1113106 between 18662-digit primes, found in 2013 by Pierre Cami, Michiel Jansen, Jens K. Andersen, PrimeForm, Primo.

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 = 1857533462 62303075693548200157509979962582683099948651188109
x65 = 871034903388863 43449123705322656962705040008760706629856986802283
x72 = 9763808660682882543001 74899353741982908376159786126663199157748266010387
x76 = 62401416110073076224658890 25426185177074468140120944390087327315890659848721
x77 = 545382416838875826681897035 90110659057865934764604873840781923513421103495579
x87 = 2798725096345871863320391354140463307 28180994209092523040703520843811319320930380677867
x92 = 438040346440298933257177107099655999301014 79007432825862362446333961919524977985103251510661
x97 = 10895334312470593108757803789229577329080364929 93138195385213105561742150447308967213141717486151
x99 = 1587947096180742294099874161743869457283715235904 52459863667791687440944143462160821328735143564091
x133 = 220505805098423836819764228021381 62423560001422663132373219086276807191581375856677 50659950011921258708497882049342113679838524318399
x155 = 573501051193549030504900787301835825160485101511983854806 08646918133804223879167823802443758585361919599047776527963058 419047009660578164772858363185263809
x156 = 151188159161413910473224635599091717347191347475741199123 99948782786332349896912875988237203296974047305051875778838643 2708639912852688724160336908656571679
x177 = 248780094097472243404896912 18520204523173891494265205671857771054034765718993 39316926569630394295966974033459801501769756510157 97243607875744813349096932598456481621971387004081
x218 = 171031486434646548 41085921840768555649752347021803947976834431221615 89881975347244280715286678546006913815669553432598 25370587691705279908930164882729553329360789780171 44057593234210769778622909436276234673417088036739
x253 = 161 75992989053204713048025383565873984999798362551566 71030473751281181199911312259550734373874520536148 51930092432794750767474667985881678018247872443196 65878436724087733884457881427402743296218118798273 49575247851843514012399313201211101277175684636727
x272 = 5664766322442249254371 23009150791462595479377358470823545512361090676515 83092270909151705930326496339047403010122690110808 76519972617985367127522222896301723496664731600712 61054435465377131309204300176522603140220490076165 04227665344802771835420837602700201417124972030829
x371 = 211456793391168128047 27903834419904799923796285605667963544243578387550 83647935493830969086740559072320624021043028003098 50677157870519355607544687390457996561399556465527 84395868077816314246284874656753669560651479732223 27951284073363128838151633809148759244135745954589 06498410646553072787910707215771803062864117907298 33109653930842047229740770849707899029657847082299
x498 = 116171901514301279412175400156653808026359353739 79308209759277071605382530759599738612297368246018 62175691501291746652325811673222667745653206998300 23811698249107345053466985654018556184168809284125 60366421679248491182605732144871277791632154674497 90379623708263605197999151147030623041167219366556 26192949993256134083071460901009368260670195324755 70235172501624476970650516467246984007050916898022 53524203352369034426800404210061512555668378897652 62056304737646834967091494264642282692101351482213
x632 = 29453397765450271545399188085266 55368252378620585099496385650600431498269661083903 16211042912735310776015757228962737061429256177227 59452435429488328389328281466289664367352954006161 79207095576921259775792175026579617936878099659414 32837668975308693630297479962123616982055909919099 17025496933775418577095695897932136276184735064982 12549583475520940170609152997656163627242028405951 20483292467767923352271563327090757509953181908766 84457108535835673100713235902439791043089273743933 82074800677693506138530042890392327727580262290580 36426781990814418117965800120148977404531919575260 38333320588240996195703518136355252551601080488639
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
Last updated 12 October 2019