Primes in Arithmetic Progression Records

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Last updated: 6 April 2024

Primes in Arithmetic Progression Records - PAP

Note: Original page was created by Jens Kruse Andersen (2005 - 2019).
This page is developed and maintained by Norman Luhn (since Oct 2021). Contact: pzktupel[at]pzktupel[dot]de.

History Of Additions (2005 - today)
Introduction
Submissions
Credited programs and projects
Links

k	Likely recent occurrences of the first known AP - k	n's	Digits	When	Discoverer
Table for the first known AP - k
12	23143 + 13# • n	0..11	6	1958	V. A. Golubev
13	4943 + 2 • 13# • 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 + 9 • 19# • n	0..16	10	1977	S. Weintraub
18	107928278317 + 1023 • 19# • n	0..17	12	1982	Paul A. Pritchard
19	8297644387 + 431 • 19# • n	0..18	11	1984	Paul A. Pritchard
20	214861583621 + 1943 • 19# • n	0..19	12	1987	Guy, James Fry, Jeff Young
21	142072321123 + 6364 • 23# • n	0..20	14	1990	James Fry, Jeff Young
22	11410337850553 + 20660 • 23# • n	0..21	15	17 Mar 1993	Paul A. Pritchard et al.
23	56211383760397 + 199678 • 23# • n	0..22	16	24 Jul 2004	Markus Frind, Paul Jobling & Paul Underwood
24	468395662504823 + 205619 • 23# • n	0..23	16	18 Jan 2007	Jaroslaw Wroblewski
25	6171054912832631 + 366384 • 23# • n	0..24	16	17 May 2008	Raanan Chermoni & Jaroslaw Wroblewski
26	43142746595714191 + 23681770 • 23# • n	0..25	18	12 Apr 2010	Benoät Perichon, PrimeGrid, AP26
27	224584605939537911 + 81292139 • 23# • n	0..26	18	19 Sep 2019	Rob Gahan, PrimeGrid, AP26

k	Primes	n's	Digits	Year	Discoverer	Record History
The largest known AP - k
3	(503 • 2^1092022 - 1) + (1103 • 2^3558176 - 503 • 2^1092022) • n	0..2	1071122	2022	Ryan Propper & Serge Batalov, EMsieve, OpenPFGW	click
4	(263093407 + 928724769 • n) • 2⁹⁹⁹⁰¹ - 1	0..3	30083	2022	Serge Batalov, NewPGen, OpenPFGW	click
5	(440012137 + 18195056 • n) • 30941# + 1	0..4	13338	2022	Serge Batalov	click
6	(4077098059 + 4441450082 • n) • 24499# + 1	0..5	10594	2024	Peter Kaiser	click
7	(2554152639 + 577051223 • n) • 7927# - 1	0..6	3407	2022	Serge Batalov	click
8	(48098104751 + 3026809034 • n) • 5303# + 1	0..7	2271	2019	Norman Luhn, Paul Underwood, Ken Davis, Primeform e - group, NewPGen, PrimeForm	click
9	(65502205462 + 6317280828 • n) • 2371# + 1	0..8	1014	2012	Ken Davis, Paul Underwood, PrimeForm egroup, NewPGen, PrimeForm	click
10	(20794561384 + 1638155407 • n) • 1050# + 1	0..9	450	2019	Norman Luhn, NewPGen, PrimeForm	click
11	(16533786790 + 1114209832 • n) • 666# + 1	0..10	289	2019	Norman Luhn	click
12	(15079159689 + 502608831 • n) • 420# + 1	0..11	180	2019	Norman Luhn	click
13	(50448064213 + 4237116495 • n) • 229# + 1	0..12	103	2019	Norman Luhn, NewPGen, PrimeForm	click
14	(55507616633 + 670355577 • n) • 229# + 1	0..13	103	2019	Norman Luhn, NewPGen, PrimeForm	click
15	(14512034548 + 87496195 • n) • 149# + 1	0..14	68	2019	Norman Luhn	click
16	(9276888728 + 43315006 • n) • 89# + 1	0..15	45	2023	Joris Perrenet	click
17	(7962733829 + 86177776 • n) • 89# + 1	0..16	45	2023	Joris Perrenet	click
18	(603518992 + 16503648 • n) • 61# + 1	0..17	33	2023	Joris Perrenet	click
19	(33277396902 + 139569962 • n) • 53# + 1	0..18	31	2019	Norman Luhn, NewPGen, PrimeForm	click
20	23 + 134181089232118748020 • 19# • n	0..19	29	2017	Wojciech Izykowski	click
21	5547796991585989797641 + 29# • n	0..20	22	2014	Jaroslaw Wroblewski	click
22	22231637631603420833 + 8 • 41# • n	1..22	20	2014	Jaroslaw Wroblewski	click
23	22231637631603420833 + 8 • 41# • n	0..22	20	2014	Jaroslaw Wroblewski	click
24	394752057712022771 + 315177938 • 23# • n	0..23	19	2024	Pavel Kirpichenko, PrimeGrid, AP26	click
25	290969863970949269 + 322359616 • 23# • n	0..24	19	2024	"tcd", PrimeGrid, AP26	click
26	236832297156314843 + 287316414 • 23# • n	0..25	19	2023	Brian D. Niegocki, PrimeGrid, AP26	click
27	605185576317848261 + 155368778 • 23# • n	0..26	19	2023	Michael Kwok, 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.

k	Primes	n's	Digits	Year	Discoverer
Smallest AP - k with minimal difference
3	3 + 2 • n	0..2	1
4	5 + 6 • n	0..3	2
5	5 + 6 • n	0..4	2
6	7 + 30 • n	0..5	3	1909	G. Lemaire
7	7 + 150 • n	0..6	3	1909	G. Lemaire
8	199 + 210 • n	0..7	4	1910	Edward B. Escott
9	199 + 210 • n	0..8	4	1910	Edward B. Escott
10	199 + 210 • n	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 + 9959 • 19# • n	0..21	15	2006	Markus Frind
23	403185216600637 + 9523 • 23# • n	0..22	15	2006	Markus Frind
24	158209144596158501 + 65073 • 23# • n	0..23	18	2014	Bryan Little, AP26
25	6171054912832631 + 366384 • 23# • n	0..24	16	2008	Raanan Chermoni & Jaroslaw Wroblewski
26	3486107472997423 + 1666981 • 23# • n	0..25	17	2012	James Fry
27	224584605939537911 + 81292139 • 23# • 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.

k	Primes	n's	Digits	Year	Discoverer
Smallest AP - k with minimal start
3	3 + 2 • n	0..2	1
4	5 + 6 • n	0..3	2
5	5 + 6 • n	0..4	2
6	7 + 30 • n	0..5	3	1909	G. Lemaire
7	7 + 150 • n	0..6	3	1909	G. Lemaire
8	11 + 5763 • 7# • n	0..7	7
9	11 + 155577 • 7# • n	0..8	9	1993	Siemion Fajtlowicz
10	11 + 1069022 • 7# • n	0..9	10	1999	Gennady Gusev
11	11 + 7315048 • 7# • n	0..10	11	1986	Günter Löh
12	13 + 641865320 • 11# • n	0..11	14	1994	W. Holsztynski, Micha Hofri
13	13 + 4293861989 • 11# • n	0..12	15	1986	Günter Löh
14	17 + 8858801964 • 13# • n	0..13	16	2001	Gennady Gusev
15	17 + 8858801964 • 13# • n	0..14	16	2001	Gennady Gusev
16	17 + 378230305161714 • 13# • n	0..15	21	2005	Phil Carmody
17	17 + 11387819007325752 • 13# • n	0..16	22	2001	Phil Carmody
18	19 + 251988718036418903 • 17# • n	0..17	25	2012	Gennady Gusev
19	19 + 4244193265542951705 • 17# • n	0..18	26	2013	Wojciech Izykowski
20	23 + 134181089232118748020 • 19# • n	0..19	29	2017	Wojciech Izykowski
21	124701216737 + 9986827 • 19# • n	0..20	16	2009	Ryszard Walczak from BOINC@Poland, Jaroslaw Wroblewski
22	1322554958713 + 2861998 • 23# • n	0..21	17	2009	Jacek Kotnowski, PrimeGrid, AP26
23	18381846925451 + 202602444 • 23# • n	0..22	19	2020	Grebuloner, PrimeGrid AP26
24	18381846925451 + 202602444 • 23# • n	0..23	19	2020	Grebuloner, PrimeGrid AP26
25	2648861307187097 + 94293751 • 23# • n	0..24	18	2018	Tom Greer, PrimeGrid, AP26
26	3486107472997423 + 1666981 • 23# • n	0..25	17	2012	James Fry
27	224584605939537911 + 81292139 • 23# • 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".

k	Primes	n's	Digits	Year	Discoverer
AP - k with minimal end
3	3 + 2 • n (7)	0..2	1
4	5 + 6 • n (23)	0..3	2
5	5 + 6 • n (29)	0..4	2
6	7 + 30 • n (157)	0..5	3	1909	G. Lemaire
7	7 + 150 • n (907)	0..6	3	1909	G. Lemaire
8	199 + 210 • n (1669)	0..7	4	1910	Edward B. Escott
9	199 + 210 • n (1879)	0..8	4	1910	Edward B. Escott
10	199 + 210 • n (2089)	0..9	4	1910	Edward B. Escott
11	110437 + 6 • 11# • n (249037)	0..10	6	1967	Edgar Karst
12	110437 + 6 • 11# • n (262897)	0..11	6	1967	Edgar Karst
13	4943 + 2 • 13# • n (725663)	0..12	6	1963	V. N. Seredinskij
14	31385539 + 14 • 13# • n (36850999)	0..13	8	1983	Paul Pritchard
15	115453391 + 138 • 13# • n (173471351)	0..14	9	1983	Paul Pritchard
16	53297929 + 323 • 13# • n (198793279)	0..15	9	1976	Sol Weintraub
17	3430751869 + 171 • 17# • n (4827507229)	0..16	10	1977	Sol Weintraub
18	4808316343 + 1406 • 17# • n (17010526363)	0..17	11	1983	Paul Pritchard
19	8297644387 + 431 • 19# • n (83547839407)	0..18	11	1984	Paul Pritchard
20	214861583621 + 1943 • 19# • n (572945039351)	0..19	12	1987	Jeff Young & James Fry
21	5749146449311 + 2681 • 19# • n (6269243827111)	0..20	13	1992	Paul Pritchard
22	19261849254523 + 80910 • 19# • n (35742689530423)	0..21	14	2023	Joris Perrenet
23	403185216600637 + 9523 • 23# • n (449924511422857)	0..22	15	2006 2023	Markus Frind Joris Perrenet proved to be minimal
24	515486946529943 + 136831 • 23# • n (1217585417914253)	0..23	16	2008	Raanan Chermoni & Jaroslaw Wroblewski
25	6171054912832631 + 366384 • 23# • n (8132758706802551)	0..24	16	2008	Raanan Chermoni & Jaroslaw Wroblewski
26	3486107472997423 + 1666981 • 23# • n (12783396861134173)	0..25	17	2012	James Fry
27	224584605939537911 + 81292139 • 23# • n (696112717486210091)	0..26	18	2019	Rob Gahan, PrimeGrid, AP26
Record history for smallest AP - k with minimal end

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 + d • n 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 + d • n, 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.

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 a • q# + 1 for fixed q. Then find AP's among the a's.
This gives an AP formula like (4941928071 + 176836494 • n) • 2411# + 1, for n=0..7
It can be written on the form p + d • n: (4941928071 • 2411# + 1) + (176836494 • 2411#) • 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.

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 Penné.
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)

Some of the data on minimal difference/start/end AP - k supplied by Gennady Gusev.