Primes in Arithmetic Progression Records

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

Last updated: 28 November 2021

News
2021
November 25. AP24 with minimaler start, 14 digits by Grebuloner, PrimeGrid AP27 ( 14 Nov 2020 )
November 25. AP25 with minimaler start, 16 digits by tng, PrimeGrid, AP26 ( 29 Dec 2018 )
October 16. AP26 (also AP25, AP24) record with 19 digits by Tuna Ertemalp, PrimeGrid, AP26
October 16. AP26 (also AP25, AP24) record with 19 digits by Rusty Clark, PrimeGrid, AP26
October 16. AP26 (also AP25, AP24) record with 18 digits by Glenn Hall, PrimeGrid, AP26
October 16. AP26 (also AP25, AP24) record with 18 digits by Brian D. Niegocki, PrimeGrid, AP26
September 24. AP3 record with 884748 digits by Serge Batalov
September 23. AP3 record with 807954 digits by Serge Batalov
September 07. AP6 and AP5 record with 10593 digits by Peter Kaiser

History Of Additions (2004 - 2019)

Introduction
Submissions

Table for first known AP-k

The largest known AP-k

Smallest AP-k with minimal difference
  Record history for k = 18 to 27

Smallest AP-k with minimal start
  Record history for k = 16 to 27

AP-k with minimal end
  Record history

Credited programs and projects

Links

Introduction
This page shows the largest known case of k primes in AP (arithmetic progression) for each k, with a record history. It also shows the minimal or smallest known difference, start and end in an AP-k, with record histories. Finally it shows all known AP24 until an AP26 was found, and all known AP25. Record histories begin with the record when the page opened in January 2005.
An AP-k is k primes of the form p + dn for some d (the difference between the primes) and k consecutive values of n. Technically any 2 primes form an AP2 but this page is only about AP3 and longer. Some sources say PAP-k instead of AP-k to signal it is primes in AP.
The Largest Known CPAP's is a different page with records for consecutive primes in AP.

Dirichlet's Theorem on Primes in Arithmetic Progressions says there are always infinitely many primes of the form c + dn, when c and d are relatively prime - and consecutive n's are not demanded.
Ben Green & Terence Tao presented a proof in 2004 that The primes contain arbitrarily long arithmetic progressions. It shows existence of AP's but does not help in finding them.

Table for the first known AP-k
k Likely recent occurrences of the first known AP-k n's Digits When? Discoverer
12
23143 + 13#n 0..11 6 1958 V. A. Golubev
13
4943 + 213#n 0..12 6 1963 V. N. Seredinskij
14
2236133941 + 23#n 0..13 10 1969 S. C. Root
15
2236133941 + 23#n 0..14 10 1969 S. C. Root
16
2236133941 + 23#n 0..15 10 1969 S. C. Root
17
3430751869 + 919#n 0..16 10 1977 S. Weintraub
18
107928278317 + 102319#n 0..17 12 1982 Paul A. Pritchard
19
8297644387 + 43119#n 0..18 11 1984 Paul A. Pritchard
20
214861583621 + 194319#n 0..19 12 1987 Guy, James Fry, Jeff Young
21
142072321123 + 636423#n 0..20 14 1990 James Fry, Jeff Young
22
11410337850553 + 2066023#n 0..21 15 17 Mar 1993 Paul A. Pritchard et al.
23
56211383760397 + 19967823#n 0..22 16 24 Jul 2004 Markus Frind, Paul Jobling & Paul Underwood
24
468395662504823 + 20561923#n 0..23 16 18 Jan 2007 Jaroslaw Wroblewski
25
6171054912832631 + 36638423#n 0..24 16 17 May 2008 Raanan Chermoni & Jaroslaw Wroblewski
26
43142746595714191 + 2368177023#n 0..25 18 12 Apr 2010 Benot Perichon, PrimeGrid, AP26
27
224584605939537911 + 8129213923#n 0..26 18 19 Sep 2019 Rob Gahan, PrimeGrid, AP26


k# (called k primorial) is the product of all primes ≤ k, e.g. 10# = 2 3 5 7.
2# = 2, 3# = 6, 5# =30, 7# = 210, 11# = 2310, 13# = 30030, 17# = 510510, 19# = 9699690, 23# = 223092870.
Unless k is prime and starts an AP-k, the difference in an AP-k is always a multiple of k# to avoid factors ≤ k. Expressions with larger primorials are often used in AP searches to avoid more small factors. Most large AP's are found roughly like this:
First compute a large set of primes aq#+1 for fixed q. Then find AP's among the a's.
This gives an AP formula like (4941928071 + 176836494n)2411#+1, for n=0..7
It can be written on the form p + dn: (49419280712411#+1) + (1768364942411#)n

Submissions
I would like to hear of all new records. Please mail any you find or know about to jens.k.a@get2net.dk. Say who should be credited, and which program proved the primes if they are above 500 digits.
All primes must be proven, i.e. prp's (probable primes) are not allowed.
The largest prime in an AP determines which AP is judged largest, so an AP3 record could start at 3.
An AP-k is also considered to be an AP-(k-1), so it can hold the record for different k's.

n's not starting at 0 indicates the end of a longer AP. The listed number of digits is for the largest prime in all tables. A year link is to an announcement. The k value links to the record history.

The largest known AP-k
k Primes n's Digits Year Discoverer Record History
3(5606879602425  21290000-1) + (3322939064-560687960242521290000)n 0..2 884748 2021 Serge Batalov
click
4 (1021747532 + 7399459n)60013#+1 0..3 25992 2019 Ken Davis, NewPGen, PrimeForm
click
5 (2738129976 + 56497325n)24499#+1 1..5 10593 2021 Peter Kaiser
click
6 (2738129976 + 56497325n)24499#+1 0..5 10593 2021 Peter Kaiser
click
7 (234043271 + 481789017n)7001#+1 0..6 3019 2012 Ken Davis, NewPGen, PrimeForm
click
8 (48098104751 + 3026809034n)5303#+1 0..7 2271 2019 Norman Luhn, Paul Underwood, Ken Davis,
Primeform e-group, NewPGen, PrimeForm
click
9 (65502205462 + 6317280828n)2371#+1 0..8 1014 2012 Ken Davis, Paul Underwood, PrimeForm
egroup, NewPGen, PrimeForm
click
10 (20794561384 + 1638155407n)1050#+1 0..9 450
2019 Norman Luhn, NewPGen, PrimeForm
click
11 (16533786790 + 1114209832n)666#+1 0..10 289 2019 Norman Luhn
click
12 (15079159689 + 502608831n)420#+1 0..11 180 2019 Norman Luhn
click
13 (50448064213 + 4237116495n)229#+1 0..12 103
2019 Norman Luhn, NewPGen, PrimeForm
click
14 (55507616633 + 670355577n)229#+1 0..13 103
2019 Norman Luhn, NewPGen, PrimeForm
click
15 (14512034548 + 87496195n)149#+1
0..14 68 2019 Norman Luhn
click
16 (9700128038 + 75782144n)83#+1 1..16 43 2019 Norman Luhn
click
17 (9700128038 + 75782144n)83#+1 0..16 43 2019Norman Luhn
click
18 (33277396902 + 139569962n)53#+1 1..18 31 2019 Norman Luhn, NewPGen, PrimeForm
click
19 (33277396902 + 139569962n)53#+1 0..18 31 2019 Norman Luhn, NewPGen, PrimeForm
click
20 23 + 13418108923211874802019#n 0..19 29 2017 Wojciech Izykowski
click
21 5547796991585989797641 + 29#n 0..20 22 2014 Jaroslaw Wroblewski
click
22 22231637631603420833 + 841#n 1..22 20 2014 Jaroslaw Wroblewski
click
23 22231637631603420833 + 841#n 0..22 20 2014 Jaroslaw Wroblewski
click
24 260947961525929049 + 16614365423#n 2..25 19 2021 Tuna Ertemalp, PrimeGrid, AP26
click
25 260947961525929049 + 16614365423#n 1..25 19 2021 Tuna Ertemalp, PrimeGrid, AP26
click
26 260947961525929049 + 16614365423#n 0..25 19 2021 Tuna Ertemalp, PrimeGrid, AP26
click
27 224584605939537911 + 8129213923#n 0..26 18 2019 Rob Gahan, PrimeGrid, AP26
click

The minimal possible difference in an AP-k is conjectured to be k# for all k > 7, proved for k ≤ 21 as of 2013. The smallest starting prime for the minimal difference is shown below. Finding an AP with a predetermined difference is much harder than an arbitrary difference. Red entries are the smallest known difference when the minimal has not been found.

Smallest AP-k with minimal difference
k Primes n's Digits Year Discoverer
3 3 + 2n 0..2 1    
4 5 + 6n 0..3 2    
5 5 + 6n 0..4 2    
6 7 + 30n 0..5 3 1909 G. Lemaire
7 7 + 150n 0..6 3 1909 G. Lemaire
8 199 + 210n 0..7 4 1910 Edward B. Escott
9 199 + 210n 0..8 4 1910 Edward B. Escott
10 199 + 210n 0..9 4 1910 Edward B. Escott
11 60858179 + 11#n 0..10  8 1999 David W. Wilson
12 147692845283 + 11#n 0..11 12 1999 David W. Wilson
13 14933623 + 13#n 0..12 8 1999 David W. Wilson
14 834172298383 + 13#n 0..13 12 2004 Gennady Gusev
15 894476585908771 + 13#n 0..14 15 2004 Jens Kruse Andersen
16 1275290173428391 + 13#n 0..15 16 2004 Gennady Gusev & Jens Kruse Andersen
17 259268961766921 + 17#n 0..16  15 2004 Gennady Gusev & Jens Kruse Andersen
18 1027994118833642281 + 17#n 0..17 19 2005 Gennady Gusev & Jens Kruse Andersen
19 1424014323012131633 + 19#n 0..18 19 2008 Jaroslaw Wroblewski
20 1424014323012131633 + 19#n 0..19 19 2008 Jaroslaw Wroblewski
21 28112131522731197609 + 19#n 0..20 20 2008 Jaroslaw Wroblewski
22 166537312120867 + 995919#n 0..21 15 2006 Markus Frind
23 403185216600637 + 952323#n 0..22 15 2006 Markus Frind
24 158209144596158501 + 6507323#n 0..23 18 2014 Bryan Little, AP26
25 6171054912832631 + 36638423#n 0..24 16 2008 Raanan Chermoni & Jaroslaw Wroblewski
26 3486107472997423 + 166698123#n 0..25 17 2012 James Fry
27 224584605939537911 + 8129213923#n 0..26 18 2019 Rob Gahan, PrimeGrid, AP26

Record history for smallest AP-k with minimal difference

The minimal starting prime in an AP-k is conjectured to be the smallest prime ≥ k, proved for k ≤ 20 as of 2017. Smaller start primes would lead to a number with a factor ≤ k. The smallest difference for the minimal start is shown below. There may have been earlier discoverers in some cases. Red primes are the smallest known when the minimal is unknown.

Smallest AP-k with minimal start
k Primes n's Digits Year Discoverer
3 3 + 2n 0..2 1    
4 5 + 6n 0..3 2    
5 5 + 6n 0..4 2    
6 7 + 30n 0..5 3 1909 G. Lemaire
7 7 + 150n 0..6 3 1909 G. Lemaire
8 11 + 57637#n 0..7 7    
9 11 + 1555777#n 0..8 9 1993 Siemion Fajtlowicz
10 11 + 10690227#n 0..9 10 1999 Gennady Gusev
11 11 + 73150487#n 0..10 11 1986 Gnter Lh
12 13 + 64186532011#n 0..11 14 1994 W. Holsztynski, Micha Hofri
13 13 + 429386198911#n 0..12 15 1986 Gnter Lh
14 17 + 885880196413#n 0..13 16 2001 Gennady Gusev
15 17 + 885880196413#n 0..14 16 2001 Gennady Gusev
16 17 + 37823030516171413#n 0..15 21 2005 Phil Carmody
17 17 + 1138781900732575213#n 0..16 22 2001 Phil Carmody
18 19 + 25198871803641890317#n 0..17 25 2012 Gennady Gusev
19 19 + 424419326554295170517#n 0..18 26 2013 Wojciech Izykowski
20 23 + 13418108923211874802019#n 0..19 29
2017 Wojciech Izykowski
21 124701216737 + 998682719#n 0..20 16 2009 Ryszard Walczak from BOINC@Poland, Jaroslaw Wroblewski
22 1322554958713 + 286199823#n 0..21 17 2009 Jacek Kotnowski, PrimeGrid, AP26
23 20389023122473 + 578554623#n 0..22 17 2009 Eric Markle, PrimeGrid, AP26
24 18381846925451 + 20260244423#n 0..23 19 2020 Grebuloner, PrimeGrid AP27
25 2648861307187097 + 9429375123#n 0..24 182018 tng, PrimeGrid, AP26
26 3486107472997423 + 166698123#n 0..25 17 2012 James Fry
27 224584605939537911 + 8129213923#n 0..26 18 2019 Rob Gahan, PrimeGrid, AP26

Record history for smallest AP-k with minimal start

Unlike the minimal difference and starting prime, there is probably no system in the minimal ending prime (shown below in parentheses) in an AP-k. Red primes are the smallest known when the minimal is unknown. It appears unlikely there exists a value of k with two AP-k sharing the minimal end, so the table heading does not begin with "Smallest".

AP-k with minimal end
k Primes n's Digits Year Discoverer
3 3 + 2n (7) 0..2 1    
4 5 + 6n (23) 0..3 2    
5 5 + 6n (29) 0..4 2    
6 7 + 30n (157) 0..5 3 1909 G. Lemaire
7 7 + 150n (907) 0..6 3 1909 G. Lemaire
8 199 + 210n (1669) 0..7 4 1910 Edward B. Escott
9 199 + 210n (1879) 0..8 4 1910 Edward B. Escott
10 199 + 210n (2089) 0..9 4 1910 Edward B. Escott
11 110437 + 611#n (249037) 0..10 6 1967 Edgar Karst
12 110437 + 611#n (262897) 0..11 6 1967 Edgar Karst
13 4943 + 213#n (725663) 0..12 6 1963 V. N. Seredinskij
14 31385539 + 1413#n (36850999) 0..13 8 1983 Paul Pritchard
15 115453391 + 13813#n (173471351) 0..14 9 1983 Paul Pritchard
16 53297929 + 32313#n (198793279) 0..15 9 1976 Sol Weintraub
17 3430751869 + 17117#n (4827507229) 0..16 10 1977 Sol Weintraub
18 4808316343 + 140617#n (17010526363) 0..17 11 1983 Paul Pritchard
19 8297644387 + 43119#n (83547839407) 0..18 11 1984 Paul Pritchard
20 214861583621 + 194319#n (572945039351) 0..19 12 1987 Jeff Young & James Fry
21 5749146449311 + 268119#n (6269243827111) 0..20 13 1992 Paul Pritchard
22 11410337850553 + 47518019#n (108201410428753) 0..21 15 1993 Paul Pritchard et al.
23 403185216600637 + 952323#n (449924511422857) 0..22 15 2006 Markus Frind
24 515486946529943 + 13683123#n (1217585417914253) 0..23 16 2008 Raanan Chermoni & Jaroslaw Wroblewski
25 6171054912832631 + 36638423#n (8132758706802551) 0..24 16 2008 Raanan Chermoni & Jaroslaw Wroblewski
26 3486107472997423 + 166698123#n (12783396861134173) 0..25 17 2012 James Fry
27 224584605939537911 + 8129213923#n (696112717486210091) 0..26 18 2019 Rob Gahan, PrimeGrid, AP26

Record history for smallest AP-k with minimal end

Credited programs and projects
Programs:
NewPGen by Paul Jobling.
APSieve by Michael Bell and Jim Fougeron.
APTreeSieve by Jens Kruse Andersen.
srsieve by Geoffrey Reynolds.
TwinGen by David Underbakke.
PSieve by Geoff Reynolds and Ken Brazier.
FermFact by Jim Fougeron.
LLR by Jean Penn.
PrimeForm by the OpenPFGW group with George Woltman.
Proth.exe by Yves Gallot.
PRP by George Woltman.
AP26 by Jaroslaw Wroblewski and Geoff Reynolds.
The primality proving program is only credited above 500 digits.

Projects:
BOINC@Poland (in Polish)
PrimeGrid
Riesel Prime Search
Primeform e-group


Links
From Chris Caldwell's The Prime Pages:
  The Top Twenty: Arithmetic Progressions of Primes (only primes with at least 1000 digits)
  The Prime Glossary: arithmetic sequence
  The Prime Database: Search on Arithmetic
  Dirichlet's Theorem on Primes in Arithmetic Progressions

Records for special AP's:
  Neil Sloane's On-Line Encyclopedia of Integer Sequences: A033189 (start of smallest AP-k with minimal difference), A061558 (smallest difference for AP-k with smallest start), A005115 (smallest end of an AP-k),
  Carlos Rivera's The Prime Puzzles & Problems Connection: Puzzle 34. Prime Triplets in arithmetic progression (largest AP3 starting at 3), and Puzzle 269. 13 primes in A.P. (smallest AP-k starting at k)
  Jens Kruse Andersen: The Largest Known CPAP's (consecutive primes in AP)

Other links:
  Markus Frind: 23 primes in arithmetic progression
  Ben Green & Terence Tao: The primes contain arbitrarily long arithmetic progressions
  Eric Weisstein's MathWorld: Prime Arithmetic Progression
  MathWorld about Green & Tao: Arbitrarily Long Progressions of Primes
  Adrian Chow Ho Yin: Page in chinese I cannot read (may show the first discovered AP12 to AP21)
  Wikipedia: Primes in arithmetic progression
  PrimeGrid: AP26 Search (a distributed project to find the longest AP)

This page is at http://primerecords.dk/aprecords.htm and licensed under the GFDL.
Created and maintained by Jens Kruse Andersen, jens.k.a@get2net.dk   home
Some of the data on minimal difference/start/end AP-k supplied by Gennady Gusev.

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