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.
January 18 2007 Jaroslaw Wroblewski discovered the first
known AP24 with 75
computers: 468395662504823 + 205619·23#·n, for n = 0..23.
Before that the longest known AP was 23 primes in arithmetic
progression by Markus Frind, Paul Jobling & Paul Underwood in 2004.
May 17 2008 Raanan Chermoni & Jaroslaw Wroblewski discovered the first known AP25
with a
lot of computing power: 6171054912832631 + 366384·23#·n, for n = 0..24.
April 12 2010 Benoãt Perichon and PrimeGrid discovered the first known AP26
with the distributed AP26 Search:
43142746595714191 + 23681770·23#·n, for n = 0..25.
September 23 2019 Rob Gahan and PrimeGrid discovered the first known AP27
with the AP26 program:
224584605939537911 + 81292139·23#·n, for n = 0..26.
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
News
2019
December 27. AP14 record with 103 digits by Norman Luhn, NewPGen, PrimeForm.
December 27. A second AP13 record with 103 digits by Norman Luhn, NewPGen, PrimeForm.
September 23. First known AP27 by Rob Gahan, PrimeGrid, AP26. It has 18 digits and is also the largest known AP24, AP25 and AP26.
August 28. AP13 record with 103 digits by Norman Luhn, NewPGen, PrimeForm.
August 25. AP10 record with 450 digits by Norman Luhn, NewPGen, PrimeForm.
August 11. AP8 record with 2271 digits by Norman Luhn, Paul Underwood, Ken Davis, Primeform e-group, NewPGen, PrimeForm.
July 27. AP19 (also AP18) record with 31 digits by Norman Luhn.
July 24. AP18 record with 31 digits by Norman Luhn.
July 4. A second AP4 record with 25992 digits by Ken Davis, NewPGen, PrimeForm.
June 21. A second AP18 record with 30 digits by Norman Luhn.
June 18. AP8 record with 1101 digits by Mike Oakes, NewPGen, PrimeForm.
June 18. AP11 record with 289 digits by Norman Luhn.
June 8. AP12 record with 180 digits by Norman Luhn.
May 23. AP15 record with 68 digits by Norman Luhn.
May 14. AP10 record with 340 digits by Norman Luhn.
April 20. AP15 record with 60 digits by Mike Oakes.
March 24. AP10 record with 327 digits by Mike Oakes.
March 8. AP4 record with 25992 digits by Ken Davis, NewPGen, PrimeForm.
February 28. AP13 record with 101 digits by Norman Luhn.
February 25. AP18 record with 30 digits by Norman Luhn.
February 20. AP17 (also AP16) record with 43 digits by Norman Luhn.
February 19. AP15 record with 57 digits by Norman Luhn.
January 25. AP14 record with 74 digits by Norman Luhn.
January 19. AP10 record with 300 digits by Norman Luhn and Gerd Lamprecht.
2018
November 1. AP5 record with 10378 digits by Ken Davis, NewPGen, PrimeForm.
August 12. AP26 (also AP25 and AP24) record with 18 digits by Syracuse University, PrimeGrid, AP26.
July 19. AP5 record with 10377 digits by Ken Davis, NewPGen, PrimeForm.
May 5. AP13 record with 93 digits by Norman Luhn, NewPGen, PrimeForm.
April 22. AP6 record with 7036 digits by Ken Davis, NewPGen, PrimeForm.
2017
December 1. First known AP20 with the minimal
start 23 found by Wojciech Izykowski. Also AP20 record (largest known
end) with 29 digits.
November 10. AP25 (also AP24) record with 18 digits by Bill Brandt-Gasuen, PrimeGrid, AP26.
August 21. AP25 (also AP24) record with 18 digits by David E. Miller, PrimeGrid, AP26.
August 19. AP6 record with 3343 digits by Michael Stocker, PrimeForm.
June 22. AP5 record with 10114 digits by Michael Stocker, NewPGen, PrimeForm.
June 17. AP25 (also AP24) record with 18 digits by William Donovan, PrimeGrid, AP26.
March 25. AP5 record with 7657 digits by Michael Stocker, NewPGen, PrimeForm.
2016
December 18. AP3 record with 435054 digits by David Broadhurst, NewPGen, PrimeForm, et al.
June 17. AP5 record with 7035 digits by Ken Davis, NewPGen, PrimeForm.
June 12. AP3 record with 388342 digits by Serge Batalov, NewPGen, LLR, et al.
April 1. A fifth AP24 record with 18 digits by Bryan Little, AP26.
January 21. AP3 record with 388342 digits by Serge Batalov, NewPGen, LLR, et al.
2015
December 4. AP5 record with 7026 digits by Serge Batalov, NewPGen, PrimeForm.
September 29. AP3 record with 388342 digits by David
Broadhurst, NewPGen, PrimeForm, et al.
February 20. A second AP26 (also AP25 and AP24) record with 18 digits by Bryan Little, AP26.
2014
September 2. AP17 (also AP16) record with 42 digits by Jaroslaw
Wroblewski.
July 9. A third AP25 record with 18 digits by Bryan Little, AP26.
June 24. AP24 with smallest known difference (65073·23#) by Bryan Little,
AP26.
June 11. This page is moved to http://primerecords.dk/aprecords.htm.
May 25. A third AP24 record with 18 digits by Bryan Little, AP26.
April 25. AP20 record with 25 digits by Jaroslaw
Wroblewski.
April 16. AP23 (also AP22) record with 20 digits by Jaroslaw Wroblewski.
April 13. AP19 (also AP18) record with 30 digits by Jaroslaw Wroblewski.
April 2. AP21 record with 22 digits by Jaroslaw Wroblewski.
March 14. A fourth AP21 record with 20 digits by Jaroslaw Wroblewski.
March 12. A third AP21 record with 20 digits by Jaroslaw Wroblewski.
March 7. AP20 record with 23 digits by Jaroslaw Wroblewski.
February 23. AP26 (also AP25 and AP24) record with 18 digits by Bryan Little, AP26.
January 13. AP24 record with 18 digits by Bryan Little, AP26.
2013
October 17. AP25 record with 18 digits by Bryan Little, AP26.
April 5. Smallest AP19 with the minimal start 19 found by Wojciech Izykowski.
April 10 he completed showing it is the smallest.
February 26. First known AP19 with the minimal start 19 found by Wojciech
Izykowski. The difference (13234551541698967679·17#) is not the smallest
possible.
2012
October 31. Smallest AP18 with the minimal start 19 found by Gennady Gusev.
March 16. AP26 with smallest known difference (1666981·23#), start (3486107472997423)
and end (12783396861134173) by James Fry. It is the second known AP26.
February 8. AP7 (also AP6) record with 3019 digits by Ken Davis, NewPGen, PrimeForm.
January 3. A second AP9 record with 1014 digits by
Ken Davis, Paul Underwood, PrimeForm egroup, NewPGen, PrimeForm.
2011
August 1. AP9 record with 1014 digits (the first known titanic AP9) by
Ken Davis, Paul Underwood, PrimeForm egroup, NewPGen, PrimeForm.
June 8. A second AP6 record with 3019 digits by Ken Davis, NewPGen, PrimeForm.
March 1. AP3 record with 200701 digits by David Broadhurst, NewPGen, PrimeForm, et al.
2010
December 22. AP13 record with 85 digits by Mike Oakes, NewPGen, PrimeForm.
October 24. AP4 record with 24092 digits by David Broadhurst, NewPGen, PrimeForm.
August 18. AP6 record with 3019 digits by Ken Davis, NewPGen, PrimeForm.
August 15. AP7 record with 1402 digits by Mike Oakes, NewPGen, PrimeForm.
July 28. AP4 record with 15004 digits by Mike Oakes, NewPGen, PrimeForm.
July 15. AP9 record with 446 digits by Mike Oakes, NewPGen, PrimeForm.
June 5. A second AP4 record with 13556 digits by David Broadhurst, NewPGen, PrimeForm.
June 3. AP4 record with 13556 digits by David Broadhurst, NewPGen, PrimeForm.
May 22. AP6 record with 3004 digits by Mike Oakes, NewPGen, PrimeForm.
May 20. AP13 record with 83 digits by Ken Davis, NewPGen, PrimeForm.
May 18. A second AP4 record with 12050 digits by David Broadhurst, NewPGen, PrimeForm.
May 17. AP4 record with 12050 digits by David Broadhurst, NewPGen, PrimeForm.
April 30. AP11 (also AP10) record with 283 digits by Ken Davis, NewPGen, PrimeForm.
April 30. AP3 record with 159382 digits by David Broadhurst, PrimeForm, et al.
April 29. AP7 record with 1335 digits by Ken Davis, NewPGen, PrimeForm.
April 20. AP3 record with 155300 digits by David Broadhurst, PrimeForm, et al.
April 12. First known AP26 by Benoãt Perichon and PrimeGrid. It has 18 digits.
April 2. AP25 record with 18 digits by Chris Wingate, PrimeGrid, AP26.
March 14. AP25 record with 18 digits by Dave Sunderland, PrimeGrid, AP26.
February 28. AP3 record with 154804 digits by David Broadhurst, PrimeForm, et al.
February 25. AP25 record with 18 digits by Keith Pattenden, PrimeGrid, AP26.
February 24. AP24 record with 18 digits by John Petterson, PrimeGrid, AP26.
February 24. AP24 record with 18 digits by Edward Gordon, PrimeGrid, AP26.
2009
December 21. AP24 record with 18 digits by Dave Mumper, PrimeGrid, AP26.
December 19. AP24 record with 18 digits by Jesper Kågefors, PrimeGrid, AP26.
December 16. AP25 (also AP24) record with 18 digits by Rune Nordbøe Skillingstad, PrimeGrid, AP26.
December 13. AP7 record with 1328 digits by Mike Oakes, NewPGen, PrimeForm.
December 11. AP14 record with 71 digits by Mike Oakes, NewPGen, PrimeForm.
December 4. AP24 record with 18 digits by Piotr M. Zalewski, PrimeGrid, AP26.
November 25. AP25 record with 17 digits by Ian Dickinson, PrimeGrid, AP26. It is the fifth known AP25.
November 18. AP6 record with 2153 digits by Ken Davis, NewPGen, PrimeForm.
November 14. AP17 (also AP16) record with 42 digits by Mike Oakes, NewPGen, PrimeForm.
November 13. AP24 record with 18 digits by Roald Solbjørg, PrimeGrid, AP26.
November 8. AP24 record with 17 digits by Marcus Rosario, PrimeGrid, AP26.
November 5. AP24 record with 17 digits by Chris Wingate, PrimeGrid, AP26.
November 1. AP24 record with 17 digits by Ian Dickinson, PrimeGrid, AP26.
October 10. AP24 record with 17 digits by Anonymous, PrimeGrid, AP26.
October 4. AP24 record with 17 digits by John S Morris III, PrimeGrid, AP26.
September 24. AP15 record with 54 digits by Mike Oakes, NewPGen, PrimeForm.
September 18. AP24 record with 17 digits by Kevin Erickson, PrimeGrid, AP26.
September 8. AP25 record with 17 digits by Keith Dale, PrimeGrid, AP26. It is the fourth known AP25.
September 2. AP25 record with 17 digits by Jochen Beck, PrimeGrid, AP26. It is the
third known
AP25.
September 1. AP10 record with 274 digits by Mike Oakes, NewPGen, PrimeForm.
August 23. AP24 record with 17 digits by Bryan Little, PrimeGrid, AP26.
August 3. AP11 record with 196 digits by Jeff Anderson-Lee, NewPGen, PrimeForm.
July 19. AP24 record with 17 digits by Jan-Cornelius Molnar, PrimeGrid, AP26.
July 13. AP6 record with 2145 digits by Mike Oakes, NewPGen, PrimeForm.
July 9. AP24 smallest known start (39421708111691) by Mark Codding, PrimeGrid, AP26.
July 4. AP24 record with 17 digits by Carsten Hartwig, PrimeGrid, AP26.
June 30. AP24 record with 17 digits by Anonymous, PrimeGrid, AP26.
June 1. AP10 record with 265 digits by Ken Davis, NewPGen, PrimeForm.
May 29. AP23 with smallest known start (20389023122473) by Eric Markle, PrimeGrid, AP26.
May 29. AP22 with smallest known start (1322554958713) by Jacek Kotnowski, PrimeGrid, AP26.
May 16. AP9 record with 425 digits by Ken Davis, NewPGen, PrimeForm.
May 6. AP24 record with 17 digits by Andreas Kobara, PrimeGrid, AP26. Also AP24 with smallest known start
(167806194923077).
April 21. AP8 record with 1057 digits by Ken Davis, NewPGen, PrimeForm.
April 9. AP25 record with 17 digits by BOINC@Poland, PrimeGrid, AP26. It is the second known
AP25 and also has the smallest known start (2960886048458003).
April 7. AP24 record with 17 digits by Pawel Stachowiak, PrimeGrid, AP26.
March 4. AP5 record with 7009 digits by Ken Davis, NewPGen, PrimeForm.
February 22. AP21 with smallest known start (124701216737) by Ryszard Walczak from BOINC@Poland, Jaroslaw Wroblewski.
February 22. Three new tables added with record histories for the smallest known
difference, start and end in an
AP-k.
February 19. AP20 with smallest known start (566547019) by Aleksander Parkitny from BOINC@Poland, Jaroslaw Wroblewski.
February 1. AP20 with smallest known start (24670154221) by Sebastian Jaworowicz
from BOINC@Poland, Jaroslaw Wroblewski.
January 26. AP19 with smallest known start (275525353) by Mariusz Szafranski from BOINC@Poland, Jaroslaw Wroblewski.
January 23. This page is moved to http://users.cybercity.dk/~dsl522332/math/aprecords.htm (moved again in 2014).
January 21. AP19 with smallest known start (275955343) by Mariusz Szafranski from BOINC@Poland, Jaroslaw Wroblewski.
January 18. Table of all known AP24 and AP25 added.
January 8. AP19 with smallest known start (1358498353) by Machloj from BOINC@Poland, Jaroslaw Wroblewski.
2008
December 3. Smallest AP21 with the minimal difference 19# by Jaroslaw Wroblewski. Also a second AP21 record with 20 digits.
December 2. Smallest AP19 and AP20 with the minimal difference 19# by Jaroslaw Wroblewski.
December 1. AP22 record with 19 digits by Jaroslaw
Wroblewski.
August 26. AP23 (also AP22) record with 19 digits by Raanan Chermoni & Jaroslaw Wroblewski.
August 10. A sixth AP22 record with 18 digits by Raanan Chermoni & Jaroslaw Wroblewski.
August 7. AP23 (also a fifth AP22) record with 18 digits by Raanan Chermoni & Jaroslaw Wroblewski.
August 4. A fourth AP22 record with 18 digits by Raanan Chermoni & Jaroslaw Wroblewski.
July 26. A third AP22 record with 18 digits by Raanan Chermoni & Jaroslaw Wroblewski.
July 3. A second AP22 record with 18 digits by Raanan Chermoni & Jaroslaw Wroblewski.
June 29. AP22 record with 18 digits by Raanan Chermoni & Jaroslaw Wroblewski.
June 24. AP5 record with 6913 digits by David Broadhurst, NewPGen, PrimeForm.
May 23. AP19 with smallest known start (2027663243) by Jaroslaw Wroblewski.
May 17. First known AP25 by Raanan Chermoni & Jaroslaw Wroblewski. It has 16 digits.
April 21. AP24 with smallest known start (401516288036293) by Raanan Chermoni & Jaroslaw Wroblewski.
April 18. AP24 with smallest known difference (136831·23#) and end
(1217585417914253) by Raanan Chermoni & Jaroslaw Wroblewski.
April 15. A fifth AP23 record with 17 digits by Raanan Chermoni & Jaroslaw Wroblewski.
April 10. A fourth AP23 record with 17 digits by Raanan Chermoni & Jaroslaw Wroblewski.
March 22. AP24 record with 17 digits by Raanan Chermoni & Jaroslaw Wroblewski.
March 19. A third AP23 record with 17 digits by Raanan Chermoni & Jaroslaw Wroblewski.
March 2. A second AP23 record with 17 digits by Raanan Chermoni & Jaroslaw Wroblewski.
February 27. AP16 record with 38 digits by Jeff Anderson-Lee, NewPGen, PrimeForm.
February 25. AP23 record with 17 digits by Raanan Chermoni &
Jaroslaw Wroblewski. Also AP23 with smallest known start
(43760869165417).
January 28. A second AP23 record with 16 digits by Raanan Chermoni & Jaroslaw Wroblewski.
January 8. A third AP4 record with 11961 digits by Ken Davis, NewPGen, PrimeForm.
2007
December 31. AP21 with smallest known difference (208·19#) by Pawel Nowak &
Jaroslaw Wroblewski.
October 5. A second AP4 record with 11961 digits by Ken Davis, NewPGen, PrimeForm.
September 26. AP4 record with 11961 digits by Ken Davis, NewPGen, PrimeForm.
August 31. AP24 (also AP23) record with 16 digits by Raanan Chermoni & Jaroslaw
Wroblewski.
August 27. AP3 record with 137514 digits by Jens Kruse Andersen, PrimeForm, et al.
August 15. AP14 record with 69 digits by Jens Kruse Andersen.
August 4. AP18 record with 29 digits by Jens Kruse Andersen.
August 3. AP17 record with 29 digits by Jens Kruse Andersen.
July 27. AP4 record with 10544 digits by Jim Fougeron, PrimeForm.
February 21. AP22 record with 18 digits by Jaroslaw Wroblewski.
February 20. AP20 record with 21 digits by Jaroslaw Wroblewski.
February 10. AP19 record (also AP18 and AP17) with 27 digits by Jaroslaw Wroblewski.
February 9. AP5 record with 5132 digits by Ken Davis, APTreeSieve, PrimeForm.
February 9. AP19 record with 21 digits by Jaroslaw Wroblewski.
February 8. AP21 (also AP20) record with 20 digits by Jaroslaw Wroblewski.
February 7. AP20 record with 19 digits by Jaroslaw Wroblewski.
February 7. A third AP20 record with 18 digits by Jaroslaw Wroblewski.
February 6. AP21 record with 18 digits by Jaroslaw Wroblewski.
February 4. A fifth AP21 record with 17 digits by Jaroslaw Wroblewski.
February 4. A second AP22 (also a fourth AP21) record with 17 digits by Jaroslaw Wroblewski.
February 4. AP22 (also a third AP21) record with 17 digits by Jaroslaw Wroblewski.
February 3. A second AP21 record with 17 digits by Jaroslaw Wroblewski.
January 31. A second AP23 record with 16 digits by Jaroslaw Wroblewski.
January 26. AP11 record with 195 digits by Jeff Anderson-Lee, NewPGen, PrimeForm.
January 18. First known AP24 (also largest AP23 with 16 digits) by Jaroslaw Wroblewski.
January 11. AP22 with smallest known start (1365385579783) by Jaroslaw Wroblewski.
January 9. AP18 record with 24 digits and also smallest known start (47) by Gennady Gusev.
January 9. AP23 record with 16 digits by Markus Frind (found June 24 2006 but unpublished).
January 6. AP18 record with 22 digits by Jaroslaw Wroblewski.
January 4. AP20 (also AP19) with smallest known difference (5·19#) by Jaroslaw Wroblewski.
January 3. AP19 with smallest known difference (11·19#) by Jaroslaw Wroblewski.
January 1. AP21 record with 17 digits by Jaroslaw Wroblewski.
2006
December 28. AP18 record with 21 digits by Jaroslaw Wroblewski.
December 28. AP9 record with 401 digits by Mike Oakes, NewPGen, PrimeForm.
December 21. A second AP20 record with 18 digits by Jaroslaw Wroblewski.
December 18. AP20 record with 18 digits by Jaroslaw Wroblewski.
December 17. AP4 record with 10004 digits by Mike Oakes, NewPGen, PrimeForm.
December 11. AP12 (also AP11) record with 173 digits by Jeff Anderson-Lee,
NewPGen, PrimeForm.
October 4. AP11 record with 150 digits by Jeff Anderson-Lee, NewPGen, PrimeForm.
September 25. AP12 (also AP11) record with 128 digits by Jeff Anderson-Lee,
NewPGen, PrimeForm.
September 24. AP14 record with 60 digits by Mike Oakes, NewPGen, PrimeForm.
September 14. AP13 record with 78 digits by Mike Oakes, NewPGen, PrimeForm.
August 7. AP10 record with 260 digits by Mike Oakes, NewPGen, PrimeForm.
June 28. AP16 record with 36 digits by Mike Oakes, NewPGen, PrimeForm.
May 8. AP22 with smallest known difference (9959·19#) by Markus Frind.
April 28. AP6 record with 1606 digits by Ken Davis, NewPGen, PrimeForm.
April 6. AP9 record with 325 digits by Jens Kruse Andersen.
April 1. Another AP22 (also AP21 this time) record with 16 digits by Markus Frind.
April 1. AP23 with smallest known difference (9523·23#) and end (449924511422857) by Markus Frind.
March 31. AP6 record with 1503 digits by Mike Oakes, NewPGen, PrimeForm.
March 25. AP12 record with 100 digits by Jens Kruse Andersen.
March 23. AP13 record with 64 digits by Jens Kruse Andersen.
March 15. AP22 record with 16 digits by Markus Frind.
March 11. AP21 record with 16 digits by Markus Frind.
February 26. AP23 with smallest known difference (156722·23#) and end
(869507044019447) by Markus Frind.
February 18. A third AP7 (also AP6) record with 1290 digits by Ken Davis,
NewPGen, PrimeForm.
February 15. A second AP7 (also AP6) record with 1290 digits by Ken Davis,
NewPGen, PrimeForm.
February 12. AP12 record with 91 digits by Norman Luhn.
February 12. AP9 record with 301 digits by Mike Oakes, APSieve, PrimeForm.
February 11. AP7 (also AP6) record with 1290 digits by Ken Davis, NewPGen,
PrimeForm.
February 10. AP9 record with 215 digits by Norman Luhn.
February 7. AP13 record with 57 digits by Norman Luhn.
February 2. AP12 record with 77 digits by Mike Oakes, PrimeForm.
2005
December 8. AP18 with smallest known start 602521111 by Gennady Gusev.
May 8. AP5 record with 5132 digits by Ken Davis, PrimeForm.
April 24. Smallest AP18 with minimal difference found by Gennady Gusev & Jens Kruse
Andersen.
February 18. AP3 record with 129240 digits by David Broadhurst, PrimeForm, et
al.
February 3. Smallest AP16 with the minimal start 17 found by Phil Carmody.
February 2. Two new tables with minimal (or smallest known) starting and ending prime in an
AP-k.
January 24. This page is published at http://hjem.get2net.dk/jka/math/aprecords.htm.
New records for AP9 to AP19 by Jens Kruse Andersen.
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.
| k | Primes | n's | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 3 | (2723880039837 · 21290000-1) + (4125·21445205-2723880039837·21290000)·n | 0..2 | 435054 | 2016 | David Broadhurst, David Abrahmi, David Metcalfe, TwinGen, NewPGen, srsieve, PSieve, Rieselprime, PrimeGrid, LLR, PrimeForm |
| 4 | (1021747532 + 7399459·n)·60013#+1 | 0..3 | 25992 | 2019 | Ken Davis, NewPGen, PrimeForm |
| 5 | (161291608 + 59874860·n)·24001#+1 | 0..4 | 10378 | 2018 | Ken Davis, NewPGen, PrimeForm |
| 6 | (1445494494 + 141836149·n)·16301#+1 | 0..5 | 7036 | 2018 | Ken Davis, NewPGen, PrimeForm |
| 7 | (234043271 + 481789017·n)·7001#+1 | 0..6 | 3019 | 2012 | Ken Davis, NewPGen, PrimeForm |
| 8 | (48098104751 + 3026809034·n)·5303#+1 | 0..7 | 2271 | 2019 | Norman Luhn, Paul Underwood, Ken Davis, Primeform e-group, NewPGen, PrimeForm |
| 9 | (65502205462 + 6317280828·n)·2371#+1 | 0..8 | 1014 | 2012 | Ken Davis, Paul Underwood, PrimeForm egroup, NewPGen, PrimeForm |
| 10 | (20794561384 + 1638155407·n)·1050#+1 | 0..9 | 450 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 11 | (16533786790 + 1114209832·n)·666#+1 | 0..10 | 289 | 2019 | Norman Luhn |
| 12 | (15079159689 + 502608831·n)·420#+1 | 0..11 | 180 | 2019 | Norman Luhn |
| 13 | (50448064213 + 4237116495·n)·229#+1 | 0..12 | 103 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 14 | (55507616633 + 670355577·n)·229#+1 | 0..13 | 103 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 15 | (14512034548 + 87496195·n)·149#+1 | 0..14 | 68 | 2019 | Norman Luhn |
| 16 | (9700128038 + 75782144·n)·83#+1 | 1..16 | 43 | 2019 | Norman Luhn |
| 17 | (9700128038 + 75782144·n)·83#+1 | 0..16 | 43 | 2019 | Norman Luhn |
| 18 | (33277396902 + 139569962·n)·53#+1 | 1..18 | 31 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 19 | (33277396902 + 139569962·n)·53#+1 | 0..18 | 31 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 20 | 23 + 134181089232118748020·19#·n | 0..19 | 29 | 2017 | Wojciech Izykowski |
| 21 | 5547796991585989797641 + 29#·n | 0..20 | 22 | 2014 | Jaroslaw Wroblewski |
| 22 | 22231637631603420833 + 8·41#·n | 1..22 | 20 | 2014 | Jaroslaw Wroblewski |
| 23 | 22231637631603420833 + 8·41#·n | 0..22 | 20 | 2014 | Jaroslaw Wroblewski |
| 24 | 224584605939537911 + 81292139·23#·n | 3..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| 25 | 224584605939537911 + 81292139·23#·n | 2..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| 26 | 224584605939537911 + 81292139·23#·n | 1..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| 27 | 224584605939537911 + 81292139·23#·n | 0..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
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
The below table contains (to the best of my knowledge) the records per
January 1, 2005, and all records set since then.
| k | Primes | n's | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 3 | (2723880039837 · 21290000-1) + (4125·21445205-2723880039837·21290000)·n | 0..2 | 435054 | 2016 | David Broadhurst, David Abrahmi, David Metcalfe, TwinGen, NewPGen, srsieve, PSieve, Rieselprime, PrimeGrid, LLR, PrimeForm |
| 3 | (731362328595 + 1986593776260·n)·21290000-1 | 0..2 | 388342 | 2016 | Serge Batalov, Randy Ready, Simon Geis, NewPGen, TwinGen, PrimeGrid, LLR |
| 3 | (907717371825 + 1312259228802·n)·21290000-1 | 0..2 | 388342 | 2016 | Serge Batalov, James Scott Brown, Simon Geis, NewPGen, TwinGen, PrimeGrid, LLR |
| 3 | (483590093385 + 1367824406910·n)·21290000-1 | 0..2 | 388342 | 2015 | David Broadhurst, Ernst Fluttert, Randall Scalise, PrimeGrid, TwinGen, LLR, NewPGen, PrimeForm |
| 3 | (8175435610410 + 32289415560495·n)·2666666-1 | 0..2 | 200701 | 2011 | David Broadhurst, Tony Gott, Ulrich Fries, PrimeGrid, TwinGen, LLR, NewPGen, PrimeForm |
| 3 | (1539·2263359+1) + (1185·2529444-1539·2263359)·n | 0..2 | 159382 | 2010 | David Broadhurst, Sascha Dinkel, Steven Harvey, FermFact, srsieve, PrimeGrid, PrimeForm, LLR |
| 3 | (907·2338934+1) + (71·2515885-907·2338934)·n | 0..2 | 155300 | 2010 | David Broadhurst, Steven Harvey, NewPGen, srsieve, PRP, PrimeForm |
| 3 | (11347·2508209-1) + (103939·2514229-11347·2508209)·n | 0..2 | 154804 | 2010 | David Broadhurst, Thomas Ritschel, Lei Zhou, Riesel Prime Search, NewPGen, srsieve, LLR, PrimeForm |
| 3 | (1769267·2340000-1) + (1061839·2456789-1769267·2340000)·n | 0..2 | 137514 | 2007 | Jens Kruse Andersen, Jiong Sun, Daniel Heuer, NewPGen, LLR, PrimeForm |
| 3 | (197·2202534-1) + (17·2429318-197·2202534)·n | 0..2 | 129240 | 2005 | David Broadhurst, Peter Grobstich, Mark Anderson, Proth.exe, LLR, PrimeForm |
| 3 | (405769059·2180009-1) + (45·2368553-405769059·2180009)·n | 0..2 | 110948 | 2003 | Jiong Sun, NewPGen, LLR, PrimeForm |
| 4 | (1021747532 + 7399459·n)·60013#+1 | 0..3 | 25992 | 2019 | Ken Davis, NewPGen, PrimeForm |
| 4 | (1006521663 + 6205834·n)·60013#+1 | 0..3 | 25992 | 2019 | Ken Davis, NewPGen, PrimeForm |
| 4 | (1631979959·225000-1) + (164196977·280000-1631979959·225000)·n | 0..3 | 24092 | 2010 | David Broadhurst, NewPGen, PrimeForm |
| 4 | (1000362700 + 2571033·n)·34687#+1 | 0..3 | 15004 | 2010 | Mike Oakes, NewPGen, PrimeForm |
| 4 | (487069965·225001-1) + (87940527·245001-487069965·225001)·n | 0..3 | 13556 | 2010 | David Broadhurst, NewPGen, PrimeForm |
| 4 | (183009063·225002-1) + (1367721·245007-183009063·225002)·n | 0..3 | 13556 | 2010 | David Broadhurst, NewPGen, PrimeForm |
| 4 | (464049135·225001-1) + (181836669·240000-464049135·225001)·n | 0..3 | 12050 | 2010 | David Broadhurst, NewPGen, PrimeForm |
| 4 | (222139713·225000-1) + (165710943·240000-222139713·225000)·n | 0..3 | 12050 | 2010 | David Broadhurst, NewPGen, PrimeForm |
| 4 | (100997770 + 3624707·n)·27751#+1 | 0..3 | 11961 | 2008 | Ken Davis, NewPGen, PrimeForm |
| 4 | (100668506 + 809963·n)·27751#+1 | 0..3 | 11961 | 2007 | Ken Davis, NewPGen, PrimeForm |
| 4 | (101056849 + 412064·n)·27751#+1 | 0..3 | 11961 | 2007 | Ken Davis, NewPGen, PrimeForm |
| 4 | (18271126 + 9548007·n)·235000+1 | 0..3 | 10544 | 2007 | Jim Fougeron, PrimeForm |
| 4 | (2024725 + 197732·n)·23159#+1 | 0..3 | 10004 | 2006 | Mike Oakes, NewPGen, PrimeForm |
| 4 | (3897833 + 19575072·n)·225000-1 | 0..3 | 7534 | 2004 | David Broadhurst, PrimeForm |
| 5 | (161291608 + 59874860·n)·24001#+1 | 0..4 | 10378 | 2018 | Ken Davis, NewPGen, PrimeForm |
| 5 | (130673984 + 22703701·n)·24001#+1 | 0..4 | 10377 | 2018 | Ken Davis, NewPGen, PrimeForm |
| 5 | (39062872 + 8829881·n)·23431#-1 | 0..4 | 10114 | 2017 | Michael Stocker, NewPGen, PrimeForm |
| 5 | (101390511 + 36694699·n)·17761#-1 | 0..4 | 7657 | 2017 | Michael Stocker, NewPGen, PrimeForm |
| 5 | (988418181 + 7078086·n)·16301#+1 | 0..4 | 7035 | 2016 | Ken Davis, NewPGen, PrimeForm |
| 5 | (43728051 + 18797279·n)·16267#-1 | 0..4 | 7026 | 2015 | Serge Batalov, NewPGen, PrimeForm |
| 5 | (82751511 + 20333209·n)·16229#+1 | 0..4 | 7009 | 2009 | Ken Davis, NewPGen, PrimeForm |
| 5 | (2799788209 + 13265760·n)·16001#/5+1 | 0..4 | 6913 | 2008 | David Broadhurst, NewPGen, PrimeForm |
| 5 | ((49077426729 + 681402540·n) · 205881·4001#/35·(205881·4001#+1) + 6) · (205881·4001#-1) + 7 | 0..4 | 5132 | 2007 | Ken Davis, APTreeSieve, PrimeForm |
| 5 | ((3433500709 + 8222924605·n) · 205881·4001#/35·(205881·4001#+1) + 6) · (205881·4001#-1) + 7 | 0..4 | 5132 | 2005 | Ken Davis, PrimeForm |
| 5 | (207727829 + 280455315·n)·5113#/35+1 | 0..4 | 2197 | 2002 | Jim Fougeron, Hans Rosenthal, PrimeForm |
| 6 | (1445494494 + 141836149·n)·16301#+1 | 0..5 | 7036 | 2018 | Ken Davis, NewPGen, PrimeForm |
| 6 | (44003657 + 12009836·n)·7759#-1 | 0..5 | 3343 | 2017 | Michael Stocker, PrimeForm |
| 6 | (234043271 + 481789017·n)·7001#+1 | 1..6 | 3019 | 2012 | Ken Davis, NewPGen, PrimeForm |
| 6 | (987502724 + 41489812·n)·7001#+1 | 0..5 | 3019 | 2011 | Ken Davis, NewPGen, PrimeForm |
| 6 | (967858482 + 19401803·n)·7001#+1 | 0..5 | 3019 | 2010 | Ken Davis, NewPGen, PrimeForm |
| 6 | (1020460444 + 21411611·n)·6977#+1 | 0..5 | 3004 | 2010 | Mike Oakes, NewPGen, PrimeForm |
| 6 | (19303382 + 41724940·n)·5011#+1 | 0..5 | 2153 | 2009 | Ken Davis, NewPGen, PrimeForm |
| 6 | (36543956 + 6632285·n)·5003#+1 | 0..5 | 2145 | 2009 | Mike Oakes, NewPGen, PrimeForm |
| 6 | (32649185 + 3884057·n)·3739#+1 | 0..5 | 1606 | 2006 | Ken Davis, NewPGen, PrimeForm |
| 6 | (27477652 + 3603821·n)·3529#+1 | 0..5 | 1503 | 2006 | Mike Oakes, NewPGen, PrimeForm |
| 6 | (143850392 + 114858412·n)·3011#+1 | 1..6 | 1290 | 2006 | Ken Davis, NewPGen, PrimeForm |
| 6 | (135477657 + 114379137·n)·3011#+1 | 1..6 | 1290 | 2006 | Ken Davis, NewPGen, PrimeForm |
| 6 | (479778674 + 1679489·n)·3011#+1 | 1..6 | 1290 | 2006 | Ken Davis, NewPGen, PrimeForm |
| 6 | (21054583 + 8440140·n)·3011#+1 | 0..5 | 1289 | 2005 | Ken Davis, NewPGen, PrimeForm |
| 6 | (17239418 + 8274192·n)·3011#+1 | 0..5 | 1289 | 2005 | Ken Davis, NewPGen, PrimeForm |
| 6 | (18015883 + 2611270·n)·3011#+1 | 0..5 | 1288 | 2004 | Ken Davis, NewPGen, PrimeForm |
| 7 | (234043271 + 481789017·n)·7001#+1 | 0..6 | 3019 | 2012 | Ken Davis, NewPGen, PrimeForm |
| 7 | (280120647 + 30017636·n)·3301#+1 | 0..6 | 1402 | 2010 | Mike Oakes, NewPGen, PrimeForm |
| 7 | (1198555389 + 5893725·n)·3121#+1 | 0..6 | 1335 | 2010 | Ken Davis, NewPGen, PrimeForm |
| 7 | (1246733996 + 35777939·n)·3109#+1 | 0..6 | 1328 | 2009 | Mike Oakes, NewPGen, PrimeForm |
| 7 | (143850392 + 114858412·n)·3011#+1 | 0..6 | 1290 | 2006 | Ken Davis, NewPGen, PrimeForm |
| 7 | (135477657 + 114379137·n)·3011#+1 | 0..6 | 1290 | 2006 | Ken Davis, NewPGen, PrimeForm |
| 7 | (479778674 + 1679489·n)·3011#+1 | 0..6 | 1290 | 2006 | Ken Davis, NewPGen, PrimeForm |
| 7 | (347855843 + 230408192·n)·3001#+1 | 0..6 | 1287 | 2004 | Jim Fougeron, Hans Rosenthal, PrimeForm |
| 8 | (48098104751 + 3026809034·n)·5303#+1 | 0..7 | 2271 | 2019 | Norman Luhn, Paul Underwood, Ken Davis, Primeform e-group, NewPGen, PrimeForm |
| 8 | (753610389 + 60355670·n)·2591#+1 | 0..7 | 1101 | 2019 | Mike Oakes, NewPGen, PrimeForm |
| 8 | (452558752 + 359463429·n)·2459#+1 | 0..7 | 1057 | 2009 | Ken Davis, NewPGen, PrimeForm |
| 8 | (4941928071 + 176836494·n)·2411#+1 | 0..7 | 1037 | 2003 | Paul Underwood, Markus Frind, PrimeForm |
| 9 | (65502205462 + 6317280828·n)·2371#+1 | 0..8 | 1014 | 2012 | Ken Davis, Paul Underwood, PrimeForm egroup, NewPGen, PrimeForm |
| 9 | (58928599133 + 1298717501·n)·2371#+1 | 0..8 | 1014 | 2011 | Ken Davis, Paul Underwood, PrimeForm egroup, NewPGen, PrimeForm |
| 9 | (758144320 + 247707761·n)·1039#+1 | 0..8 | 446 | 2010 | Mike Oakes, NewPGen, PrimeForm |
| 9 | (190556231 + 138880294·n)·997#+1 | 0..8 | 425 | 2009 | Ken Davis, NewPGen, PrimeForm |
| 9 | (805227062 + 54790161·n)·941#+1 | 0..8 | 401 | 2006 | Mike Oakes, NewPGen, PrimeForm |
| 9 | (582170448 + 7977192·n)·757#+1 | 0..8 | 325 | 2006 | Jens Kruse Andersen |
| 9 | (20475784 + 18458425·n)·701#+1 | 0..8 | 301 | 2006 | Mike Oakes, APSieve, PrimeForm |
| 9 | (105628556 + 7914782·n)·499#+1 | 0..8 | 215 | 2006 | Norman Luhn |
| 9 | (501788528 + 12970338·n)·463#+1 | 1..9 | 202 | 2005 | Jens Kruse Andersen |
| 9 | 25127352416973 · 199#·227·229 + x87 + 210n | 0..8 | 101 | 2004 | Hans Rosenthal & Jens Kruse Andersen |
| 10 | (20794561384 + 1638155407·n)·1050#+1 | 0..9 | 450 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 10 | (5375864877 + 721683·n)·800#+1 | 0..9 | 340 | 2019 | Norman Luhn |
| 10 | (42918719 + 23865878·n)·761#+1 | 0..9 | 327 | 2019 | Mike Oakes |
| 10 | (3843864037 + 35500601·n)·691#+1 | 0..9 | 300 | 2019 | Norman Luhn and Gerd Lamprecht |
| 10 | (3186700865 + 61959394·n)·653#+1 | 1..10 | 283 | 2010 | Ken Davis, NewPGen, PrimeForm |
| 10 | (565429078 + 147743546·n)·641#+1 | 0..9 | 274 | 2009 | Mike Oakes, NewPGen, PrimeForm |
| 10 | (1600616052 + 52323192·n)·617#+1 | 0..9 | 265 | 2009 | Ken Davis, NewPGen, PrimeForm |
| 10 | (1079682357 + 109393276·n)·607#+1 | 0..9 | 260 | 2006 | Mike Oakes, NewPGen, PrimeForm |
| 10 | (501788528 + 12970338·n)·463#+1 | 0..9 | 202 | 2005 | Jens Kruse Andersen |
| 10 | 507618446770482 · 193# + x77 + 210n | 0..9 | 93 | 1998 | Manfred Toplic, Harvey Dubner, Tony Forbes, Nik Lygeros, Michel Mizony, Paul Zimmermann |
| 11 | (16533786790 + 1114209832·n)·666#+1 | 0..10 | 289 | 2019 | Norman Luhn |
| 11 | (3186700865 + 61959394·n)·653#+1 | 0..10 | 283 | 2010 | Ken Davis, NewPGen, PrimeForm |
| 11 | (197477410 + 146636·n)·457#+1 | 0..10 | 196 | 2009 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 11 | (631346030 + 151515939·n)·449#+1 | 0..10 | 195 | 2007 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 11 | (1366899295 + 54290654·n)·401#+1 | 1..11 | 173 | 2006 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 11 | (1405659596 + 55680004·n)·349#+1 | 0..10 | 150 | 2006 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 11 | (1075032530 + 89982195·n)·283#+1 | 1..11 | 128 | 2006 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 11 | (262779044 + 49839897·n)·281#+1 | 0..10 | 125 | 2005 | Jens Kruse Andersen |
| 11 | 17 + 11387819007325752·13#·n | 6..16 | 22 | 2001 | Phil Carmody |
| 12 | (15079159689 + 502608831·n)·420#+1 | 0..11 | 180 | 2019 | Norman Luhn |
| 12 | (1366899295 + 54290654·n)·401#+1 | 0..11 | 173 | 2006 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 12 | (1075032530 + 89982195·n)·283#+1 | 0..11 | 128 | 2006 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 12 | (53660216 + 27243281·n)·229#+1 | 0..11 | 100 | 2006 | Jens Kruse Andersen |
| 12 | (117531238 + 1064977·n)·199#+1 | 0..11 | 91 | 2006 | Norman Luhn |
| 12 | (30150295 + 3538849·n)·173#+1 | 0..11 | 77 | 2006 | Mike Oakes, PrimeForm |
| 12 | (8986417 + 30820926·n)·157#+1 | 0..11 | 71 | 2005 | Jens Kruse Andersen |
| 12 | 17 + 11387819007325752·13#·n | 5..16 | 22 | 2001 | Phil Carmody |
| 13 | (50448064213 + 4237116495·n)·229#+1 | 0..12 | 103 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 13 | (58494782328 + 70321714·n)·229#+1 | 0..12 | 103 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 13 | (3107048086 + 2726206·n)·229#+1 | 0..12 | 101 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 13 | (58814898610 + 1121191·n)·199#+1 | 0..12 | 93 | 2018 | Norman Luhn, NewPGen, PrimeForm |
| 13 | (1296982250 + 14976848·n)·191#+1 | 0..12 | 85 | 2010 | Mike Oakes, NewPGen, PrimeForm |
| 13 | (7967748420 + 340308346·n)·181#+1 | 0..12 | 83 | 2010 | Ken Davis, NewPGen, PrimeForm |
| 13 | (1374042988 + 22886141·n)·173#+1 | 0..12 | 78 | 2006 | Mike Oakes, NewPGen, PrimeForm |
| 13 | (208988611 + 32224644·n)·139#+1 | 0..12 | 64 | 2006 | Jens Kruse Andersen |
| 13 | (11428680 + 6738753·n)·127#+1 | 0..12 | 57 | 2006 | Norman Luhn |
| 13 | (539591312 + 45914254·n)·103#+1 | 0..12 | 50 | 2005 | Jens Kruse Andersen |
| 13 | 17 + 11387819007325752·13#·n | 4..16 | 22 | 2001 | Phil Carmody |
| 14 | (55507616633 + 670355577·n)·229#+1 | 0..13 | 103 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 14 | (2032207368 + 34268355·n)·163#+1 | 0..13 | 74 | 2019 | Norman Luhn |
| 14 | (145978014 + 25313115·n)·157#+1 | 0..13 | 71 | 2009 | Mike Oakes, NewPGen, PrimeForm |
| 14 | (1067385825 + 193936257·n)·151#+1 | 0..13 | 69 | 2007 | Jens Kruse Andersen |
| 14 | (211370029 + 22263450·n)·131#+1 | 0..13 | 60 | 2006 | Mike Oakes, NewPGen, PrimeForm |
| 14 | (358766428 + 17143877·n)·101#+1 | 1..14 | 48 | 2005 | Jens Kruse Andersen |
| 14 | 17 + 11387819007325752·13#·n | 3..16 | 22 | 2001 | Phil Carmody |
| 15 | (14512034548 + 87496195·n)·149#+1 | 0..14 | 68 | 2019 | Norman Luhn |
| 15 | (829774854 + 61796110·n)·131#+1 | 0..14 | 60 | 2019 | Mike Oakes |
| 15 | (14563635080 + 9152216·n)·113#+1 | 0..14 | 57 | 2019 | Norman Luhn |
| 15 | (237375311 + 118560155·n)·109#+1 | 0..14 | 54 | 2009 | Mike Oakes, NewPGen, PrimeForm |
| 15 | (358766428 + 17143877·n)·101#+1 | 0..14 | 48 | 2005 | Jens Kruse Andersen |
| 15 | 17 + 11387819007325752·13#·n | 2..16 | 22 | 2001 | Phil Carmody |
| 16 | (9700128038 + 75782144·n)·83#+1 | 1..16 | 43 | 2019 | Norman Luhn |
| 16 | 442604220336549402080078796974991691613909 + 103#·n | 1..16 | 42 | 2014 | Jaroslaw Wroblewski |
| 16 | (263013824 + 18107251·n)·83#+1 | 1..16 | 42 | 2009 | Mike Oakes, NewPGen, PrimeForm |
| 16 | (636435342 + 49408956·n)·73#+1 | 0..15 | 38 | 2008 | Jeff Anderson-Lee, NewPGen, PrimeForm |
| 16 | (1723800454 + 529799·n)·71#+1 | 0..15 | 36 | 2006 | Mike Oakes, NewPGen, PrimeForm |
| 16 | (253456539 + 13892183·n)·43#+1 | 1..16 | 25 | 2005 | Jens Kruse Andersen |
| 16 | 17 + 11387819007325752·13#·n | 1..16 | 22 | 2001 | Phil Carmody |
| 17 | (9700128038 + 75782144·n)·83#+1 | 0..16 | 43 | 2019 | Norman Luhn |
| 17 | 442604220336549402080078796974991691613909 + 103#·n | 0..16 | 42 | 2014 | Jaroslaw Wroblewski |
| 17 | (263013824 + 18107251·n)·83#+1 | 0..16 | 42 | 2009 | Mike Oakes, NewPGen, PrimeForm |
| 17 | (1259891250 + 70154768·n)·53#+1 | 0..16 | 29 | 2007 | Jens Kruse Andersen |
| 17 | 62749659973280668140514103 + 107·61#·n | 2..18 | 27 | 2007 | Jaroslaw Wroblewski |
| 17 | (253456539 + 13892183·n)·43#+1 | 0..16 | 25 | 2005 | Jens Kruse Andersen |
| 17 | 17 + 11387819007325752·13#·n | 0..16 | 22 | 2001 | Phil Carmody |
| 18 | (33277396902 + 139569962·n)·53#+1 | 1..18 | 31 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 18 | (16013322956 + 918494804·n)·53#+1 | 0..17 | 31 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 18 | (19024732855 + 282152663·n)·53#+1 | 0..17 | 30 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 18 | (16682454839 + 24397524·n)·53#+1 | 0..17 | 30 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 18 | 1029+999 + 1806448944300798320195·19#·(n-2) | 1..18 | 30 | 2014 | Jaroslaw Wroblewski |
| 18 | (1051673535 + 32196596·n)·53#+1 | 0..17 | 29 | 2007 | Jens Kruse Andersen |
| 18 | 62749659973280668140514103 + 107·61#·n | 1..18 | 27 | 2007 | Jaroslaw Wroblewski |
| 18 | 47 + 19622534420177523·17#·n | 0..17 | 24 | 2007 | Gennady Gusev |
| 18 | 970080760458011996047 + 53#·n | 0..17 | 22 | 2007 | Jaroslaw Wroblewski |
| 18 | 398182070391807316627 + 53#·n | 0..17 | 21 | 2006 | Jaroslaw Wroblewski |
| 18 | (273972449 + 25355830·n)·31#+1 | 1..18 | 21 | 2005 | Jens Kruse Andersen |
| 18 | (1042322200 + 11598832·n)·23#+1 | 0..17 | 18 | 2002 | Markus Frind |
| 19 | (33277396902 + 139569962·n)·53#+1 | 0..18 | 31 | 2019 | Norman Luhn, NewPGen, PrimeForm |
| 19 | 1029+999 + 1806448944300798320195·19#·(n-2) | 0..18 | 30 | 2014 | Jaroslaw Wroblewski |
| 19 | 62749659973280668140514103 + 107·61#·n | 0..18 | 27 | 2007 | Jaroslaw Wroblewski |
| 19 | 254215977184797362303 + 53#·n | 0..18 | 21 | 2007 | Jaroslaw Wroblewski |
| 19 | (273972449 + 25355830·n)·31#+1 | 0..18 | 21 | 2005 | Jens Kruse Andersen |
| 19 | (669956329 + 7563194·n)·23#+1 | 1..19 | 18 | 2002 | Markus Frind |
| 20 | 23 + 134181089232118748020·19#·n | 0..19 | 29 | 2017 | Wojciech Izykowski |
| 20 | 3533531731191494525351461 + 61#·n | 0..19 | 25 | 2014 | Jaroslaw Wroblewski |
| 20 | 21816168581655686541377 + 59#·n | 0..19 | 23 | 2014 | Jaroslaw Wroblewski |
| 20 | 178284683588844176017 + 53#·n | 0..19 | 21 | 2007 | Jaroslaw Wroblewski |
| 20 | 1925228725347080393 + 47#·n | 1..20 | 20 | 2007 | Jaroslaw Wroblewski |
| 20 | 2962411312104427 + 185·41#·n | 0..19 | 19 | 2007 | Jaroslaw Wroblewski |
| 20 | 2036800114407689 + 66·41#·n | 0..19 | 18 | 2007 | Jaroslaw Wroblewski |
| 20 | 11735227242889999 + 43#·n | 0..19 | 18 | 2006 | Jaroslaw Wroblewski |
| 20 | 9372688136871853 + 43#·n | 0..19 | 18 | 2006 | Jaroslaw Wroblewski |
| 20 | (669956329 + 7563194·n)·23#+1 | 0..19 | 18 | 2002 | Markus Frind |
| 21 | 5547796991585989797641 + 29#·n | 0..20 | 22 | 2014 | Jaroslaw Wroblewski |
| 21 | 24657656407921828441 + 47#·n | 0..20 | 20 | 2014 | Jaroslaw Wroblewski |
| 21 | 23260414913723182141 + 47#·n | 0..20 | 20 | 2014 | Jaroslaw Wroblewski |
| 21 | 28112131522731197609 + 19#·n | 0..20 | 20 | 2008 | Jaroslaw Wroblewski |
| 21 | 1925228725347080393 + 47#·n | 0..20 | 20 | 2007 | Jaroslaw Wroblewski |
| 21 | 25521343898340097 + 37·41#·n | 0..20 | 18 | 2007 | Jaroslaw Wroblewski |
| 21 | 1245653020090313 + 384·37#·n | 0..20 | 17 | 2007 | Jaroslaw Wroblewski |
| 21 | 3135462279378283 + 316·37#·n | 1..21 | 17 | 2007 | Jaroslaw Wroblewski |
| 21 | 789337428437009 + 320·37#·n | 1..21 | 17 | 2007 | Jaroslaw Wroblewski |
| 21 | 620353098196661 + 84·37#·n | 0..20 | 17 | 2007 | Jaroslaw Wroblewski |
| 21 | 9358091177025961 + 11·37#·n | 0..20 | 17 | 2007 | Jaroslaw Wroblewski |
| 21 | 556904117141899 + 1105111·23#·n | 1..21 | 16 | 2006 | Markus Frind |
| 21 | 1291515762123569 + 947674·23#·n | 0..20 | 16 | 2006 | Markus Frind |
| 21 | 56211383760397 + 199678·23#·n | 2..22 | 16 | 2004 | Markus Frind, Paul Jobling, Paul Underwood |
| 22 | 22231637631603420833 + 8·41#·n | 1..22 | 20 | 2014 | Jaroslaw Wroblewski |
| 22 | 1351906725737537399 + 43#·n | 0..21 | 19 | 2008 | Jaroslaw Wroblewski |
| 22 | 117075039027693563 + 6548·37#·n | 1..22 | 19 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 22 | 110236462100072933 + 2422·37#·n | 0..21 | 18 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 22 | 39468294485409091 + 2416·37#·n | 1..22 | 18 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 22 | 89563668600668359 + 1840·37#·n | 0..21 | 18 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 22 | 5105850615934567 + 1788·37#·n | 0..21 | 18 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 22 | 126599504321565949 + 912·37#·n | 0..21 | 18 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 22 | 32392290909421589 + 905·37#·n | 0..21 | 18 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 22 | 950203555027421 + 892·37#·n | 0..21 | 18 | 2007 | Jaroslaw Wroblewski |
| 22 | 3135462279378283 + 316·37#·n | 0..21 | 17 | 2007 | Jaroslaw Wroblewski |
| 22 | 789337428437009 + 320·37#·n | 0..21 | 17 | 2007 | Jaroslaw Wroblewski |
| 22 | 556904117141899 + 1105111·23#·n | 0..21 | 16 | 2006 | Markus Frind |
| 22 | 1059297083391793 + 408270·23#·n | 0..21 | 16 | 2006 | Markus Frind |
| 22 | 56211383760397 + 199678·23#·n | 1..22 | 16 | 2004 | Markus Frind, Paul Jobling, Paul Underwood |
| 23 | 22231637631603420833 + 8·41#·n | 0..22 | 20 | 2014 | Jaroslaw Wroblewski |
| 23 | 117075039027693563 + 6548·37#·n | 0..22 | 19 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 39468294485409091 + 2416·37#·n | 0..22 | 18 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 660593947782971 + 5414270·23#·n | 0..22 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 542440132260899 + 4560607·23#·n | 0..22 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 3465600168789197 + 3405459·23#·n | 0..22 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 489357377433019 + 2701556·23#·n | 0..22 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 43760869165417 + 2339805·23#·n | 0..22 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 1172380401690583 + 1675204·23#·n | 0..22 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 1564588127269043 + 1249750·23#·n | 1..23 | 16 | 2007 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 242354932628899 + 291535·23#·n | 0..22 | 16 | 2007 | Jaroslaw Wroblewski |
| 23 | 468395662504823 + 205619·23#·n | 1..23 | 16 | 2007 | Jaroslaw Wroblewski |
| 23 | 654737613204229 + 110135·23#·n | 0..22 | 16 | 2007 | Markus Frind |
| 23 | 56211383760397 + 199678·23#·n | 0..22 | 16 | 2004 | Markus Frind, Paul Jobling, Paul Underwood |
| 24 | 224584605939537911 + 81292139·23#·n | 3..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| 24 | 271702189272825977 + 67515487·23#·n | 2..25 | 18 | 2018 | Syracuse University, PrimeGrid, AP26 |
| 24 | 293037522812241983 + 42713298·23#·n | 1..24 | 18 | 2017 | Bill Brandt-Gasuen, PrimeGrid, AP26 |
| 24 | 320382581839925153 + 36759418·23#·n | 1..24 | 18 | 2017 | David E. Miller, PrimeGrid, AP26 |
| 24 | 296950539631234873 + 32252465·23#·n | 1..24 | 18 | 2017 | William Donovan, PrimeGrid, AP26 |
| 24 | 163578055942973267 + 54221892·23#·n | 0..23 | 18 | 2016 | Bryan Little, AP26 |
| 24 | 161004359399459161 + 47715109·23#·n | 2..25 | 18 | 2015 | Bryan Little, AP26 |
| 24 | 154915016996669203 + 46919728·23#·n | 0..23 | 18 | 2014 | Bryan Little, AP26 |
| 24 | 136926916457315893 + 44121555·23#·n | 2..25 | 18 | 2014 | Bryan Little, AP26 |
| 24 | 133959376918860781 + 45008483·23#·n | 0..23 | 18 | 2014 | Bryan Little, AP26 |
| 24 | 28806475189976381 + 36028618·23#·n | 0..23 | 18 | 2010 | John Petterson, PrimeGrid, AP26 |
| 24 | 4076845864885399 + 35779587·23#·n | 0..23 | 18 | 2010 | Edward Gordon, PrimeGrid, AP26 |
| 24 | 6450318626856161 + 30430175·23#·n | 0..23 | 18 | 2009 | Dave Mumper, PrimeGrid, AP26 |
| 24 | 7448460750808573 + 27806605·23#·n | 0..23 | 18 | 2009 | Jesper Kågefors, PrimeGrid, AP26 |
| 24 | 12353443596260323 + 23793841·23#·n | 1..24 | 18 | 2009 | Rune Nordbøe Skillingstad, PrimeGrid, AP26 |
| 24 | 4911007789148401 + 20273384·23#·n | 0..23 | 18 | 2009 | Piotr M. Zalewski, PrimeGrid, AP26 |
| 24 | 4891686128805269 + 19453568·23#·n | 0..23 | 18 | 2009 | Roald Solbjørg, PrimeGrid, AP26 |
| 24 | 4687877159107031 + 18203167·23#·n | 0..23 | 17 | 2009 | Marcus Rosario, PrimeGrid, AP26 |
| 24 | 1948053460212667 + 17745794·23#·n | 0..23 | 17 | 2009 | Chris Wingate, PrimeGrid, AP26 |
| 24 | 3634080452156039 + 16981607·23#·n | 0..23 | 17 | 2009 | Ian Dickinson, PrimeGrid, AP26 |
| 24 | 4187489431145893 + 15636117·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 3193781476385167 + 15609111·23#·n | 0..23 | 17 | 2009 | John S Morris III , PrimeGrid, AP26 |
| 24 | 10307159737232191 + 14120563·23#·n | 0..23 | 17 | 2009 | Kevin Erickson, PrimeGrid, AP26 |
| 24 | 13678065943093049 + 13223804·23#·n | 0..23 | 17 | 2009 | Bryan Little, PrimeGrid, AP26 |
| 24 | 10317962076055027 + 10241601·23#·n | 0..23 | 17 | 2009 | Jan-Cornelius Molnar, PrimeGrid, AP26 |
| 24 | 13432401425380607 + 9219580·23#·n | 0..23 | 17 | 2009 | Carsten Hartwig, PrimeGrid, AP26 |
| 24 | 14992521666441877 + 8832442·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 167806194923077 + 4935146·23#·n | 0..23 | 17 | 2009 | Andreas Kobara, PrimeGrid, AP26 |
| 24 | 7960592659339799 + 2326495·23#·n | 0..23 | 17 | 2009 | Pawel Stachowiak, PrimeGrid, AP26 |
| 24 | 1606021011322579 + 3490622·23#·n | 0..23 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 1564588127269043 + 1249750·23#·n | 0..23 | 16 | 2007 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 468395662504823 + 205619·23#·n | 0..23 | 16 | 2007 | Jaroslaw Wroblewski |
| 25 | 224584605939537911 + 81292139·23#·n | 2..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| 25 | 271702189272825977 + 67515487·23#·n | 1..25 | 18 | 2018 | Syracuse University, PrimeGrid, AP26 |
| 25 | 293037522812241983 + 42713298·23#·n | 0..24 | 18 | 2017 | Bill Brandt-Gasuen, PrimeGrid, AP26 |
| 25 | 320382581839925153 + 36759418·23#·n | 0..24 | 18 | 2017 | David E. Miller, PrimeGrid, AP26 |
| 25 | 296950539631234873 + 32252465·23#·n | 0..24 | 18 | 2017 | William Donovan, PrimeGrid, AP26 |
| 25 | 161004359399459161 + 47715109·23#·n | 1..25 | 18 | 2015 | Bryan Little, AP26 |
| 25 | 136314114923771779 + 47064543·23#·n | 0..24 | 18 | 2014 | Bryan Little, AP26 |
| 25 | 136926916457315893 + 44121555·23#·n | 1..25 | 18 | 2014 | Bryan Little, AP26 |
| 25 | 65921086021334669 + 43905174·23#·n | 0..24 | 18 | 2013 | Bryan Little, AP26 |
| 25 | 18626565939034793 + 30821486·23#·n | 0..24 | 18 | 2010 | Chris Wingate, PrimeGrid, AP26 |
| 25 | 25300381597038677 + 28603610·23#·n | 0..24 | 18 | 2010 | Dave Sunderland, PrimeGrid, AP26 |
| 25 | 42592855872841649 + 19093314·23#·n | 0..24 | 18 | 2010 | Keith Pattenden, PrimeGrid, AP26 |
| 25 | 12353443596260323 + 23793841·23#·n | 0..24 | 18 | 2009 | Rune Nordbøe Skillingstad, PrimeGrid, AP26 |
| 25 | 46176957093163301 + 1109121·23#·n | 0..24 | 17 | 2009 | Ian Dickinson, PrimeGrid, AP26 |
| 25 | 18162964758258289 + 3755664·23#·n | 0..24 | 17 | 2009 | Keith Dale, PrimeGrid, AP26 |
| 25 | 20919497549238289 + 3155495·23#·n | 0..24 | 17 | 2009 | Jochen Beck, PrimeGrid, AP26 |
| 25 | 2960886048458003 + 2346233·23#·n | 0..24 | 17 | 2009 | BOINC@Poland , PrimeGrid, AP26 |
| 25 | 6171054912832631 + 366384·23#·n | 0..24 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 26 | 224584605939537911 + 81292139·23#·n | 1..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| 26 | 271702189272825977 + 67515487·23#·n | 0..25 | 18 | 2018 | Syracuse University, PrimeGrid, AP26 |
| 26 | 161004359399459161 + 47715109·23#·n | 0..25 | 18 | 2015 | Bryan Little, AP26 |
| 26 | 136926916457315893 + 44121555·23#·n | 0..25 | 18 | 2014 | Bryan Little, AP26 |
| 26 | 43142746595714191 + 23681770·23#·n | 0..25 | 18 | 2010 | Benoãt Perichon, PrimeGrid, AP26 |
| 27 | 224584605939537911 + 81292139·23#·n | 0..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
xN is an N-digit number which was used in a CPAP record but has no simple
expression.
x77 = 545382416838875826681897035 90110659057865934764604873840781923513421103495579
x87 = 2798725096345871863320391354140463307 28180994209092523040703520843811319320930380677867
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 |
|---|---|---|---|---|---|
| 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 |
| k | Primes | n's | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 18 | 1027994118833642281 + 17#·n | 0..17 | 19 | 2005 | Gennady Gusev & Jens Kruse Andersen |
| 18 | 24604783199 + 581·17#·n | 0..17 | 11 | 2002 | Gennady Gusev |
| 19 | 1424014323012131633 + 19#·n | 0..18 | 19 | 2008 | Jaroslaw Wroblewski |
| 19 | 20736372344841161 + 5·19#·n | 0..18 | 17 | 2007 | Jaroslaw Wroblewski |
| 19 | 63734273908111 + 11·19#·n | 0..18 | 14 | 2007 | Jaroslaw Wroblewski |
| 19 | 2773814832407 + 155·19#·n | 0..18 | 13 | 1992 | Paul Pritchard |
| 20 | 1424014323012131633 + 19#·n | 0..19 | 19 | 2008 | Jaroslaw Wroblewski |
| 20 | 20736372344841161 + 5·19#·n | 0..19 | 17 | 2007 | Jaroslaw Wroblewski |
| 20 | 2773814832407 + 155·19#·n | 0..19 | 13 | 1992 | Paul Pritchard |
| 21 | 28112131522731197609 + 19#·n | 0..20 | 20 | 2008 | Jaroslaw Wroblewski |
| 21 | 809504830479349 + 208·19#·n | 0..20 | 15 | 2007 | Pawel Nowak & Jaroslaw Wroblewski |
| 21 | 5749146449311 + 2681·19#·n | 0..20 | 13 | 1992 | Paul Pritchard |
| 22 | 166537312120867 + 9959·19#·n | 0..21 | 15 | 2006 | Markus Frind |
| 22 | 497003857949969 + 179998·19#·n | 0..21 | 15 | 2004 | Markus Frind |
| 23 | 403185216600637 + 9523·23#·n | 0..22 | 15 | 2006 | Markus Frind |
| 23 | 100308707032367 + 156722·23#·n | 0..22 | 15 | 2006 | Markus Frind |
| 23 | 56211383760397 + 199678·23#·n | 0..22 | 16 | 2004 | Markus Frind, Paul Jobling, Paul Underwood |
| 24 | 158209144596158501 + 65073·23#·n | 0..23 | 18 | 2014 | Bryan Little, AP26 |
| 24 | 515486946529943 + 136831·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 468395662504823 + 205619·23#·n | 0..23 | 16 | 2007 | Jaroslaw Wroblewski |
| 25 | 6171054912832631 + 366384·23#·n | 0..24 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 26 | 3486107472997423 + 1666981·23#·n | 0..25 | 17 | 2012 | James Fry |
| 26 | 43142746595714191 + 23681770·23#·n | 0..25 | 18 | 2010 | Benoãt Perichon, PrimeGrid, AP26 |
| 27 | 224584605939537911 + 81292139·23#·n | 0..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
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 |
|---|---|---|---|---|---|
| 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 | 20389023122473 + 5785546·23#·n | 0..22 | 17 | 2009 | Eric Markle, PrimeGrid, AP26 |
| 24 | 39421708111691 + 9740894·23#·n | 0..23 | 17 | 2009 | Mark Codding, PrimeGrid, AP26 |
| 25 | 2960886048458003 + 2346233·23#·n | 0..24 | 17 | 2009 | BOINC@Poland , 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 |
| k | Primes | n's | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 16 | 17 + 378230305161714·13#·n | 0..15 | 21 | 2005 | Phil Carmody |
| 16 | 17 + 11387819007325752·13#·n | 0..15 | 22 | 2001 | Phil Carmody |
| 17 | 17 + 11387819007325752·13#·n | 0..16 | 22 | 2001 | Phil Carmody |
| 18 | 19 + 251988718036418903·17#·n | 0..17 | 25 | 2012 | Gennady Gusev |
| 18 | 47 + 19622534420177523·17#·n | 0..17 | 24 | 2007 | Gennady Gusev |
| 18 | 602521111 + 96575729250·n | 0..17 | 13 | 2005 | Gennady Gusev |
| 18 | 1052724247 + 6549·17#·n | 0..17 | 11 | 2002 | Gennady Gusev |
| 19 | 19 + 4244193265542951705·17#·n | 0..18 | 26 | 2013 | Wojciech Izykowski |
| 19 | 19 + 13234551541698967679·17#·n | 0..18 | 27 | 2013 | Wojciech Izykowski |
| 19 | 275525353 + 407133393·19#·n | 0..18 | 17 | 2009 | Mariusz Szafranski from BOINC@Poland, Jaroslaw Wroblewski |
| 19 | 275955343 + 368042893·19#·n | 0..18 | 17 | 2009 | Mariusz Szafranski from BOINC@Poland, Jaroslaw Wroblewski |
| 19 | 1358498353 + 16584560·19#·n | 0..18 | 16 | 2009 | Machloj from BOINC@Poland, Jaroslaw Wroblewski |
| 19 | 2027663243 + 1002606·19#·n | 0..18 | 15 | 2008 | Jaroslaw Wroblewski |
| 19 | 8297644387 + 431·19#·n | 0..18 | 11 | 1984 | Paul Pritchard |
| 20 | 23 + 134181089232118748020·19#·n | 0..19 | 29 | 2017 | Wojciech Izykowski |
| 20 | 566547019 + 846627559·19#·n | 0..19 | 18 | 2009 | Aleksander Parkitny from BOINC@Poland, Jaroslaw Wroblewski |
| 20 | 24670154221 + 370766·19#·n | 0..19 | 14 | 2009 | Sebastian Jaworowicz from BOINC@Poland, Jaroslaw Wroblewski |
| 20 | 68469367129 + 4926·19#·n | 0..19 | 12 | 1992 | Paul Pritchard |
| 21 | 124701216737 + 9986827·19#·n | 0..20 | 16 | 2009 | Ryszard Walczak from BOINC@Poland, Jaroslaw Wroblewski |
| 21 | 142072321123 + 146372·19#·n | 0..20 | 14 | 1990 | Paul Pritchard |
| 22 | 1322554958713 + 2861998·23#·n | 0..21 | 17 | 2009 | Jacek Kotnowski, PrimeGrid, AP26 |
| 22 | 1365385579783 + 405927·23#·n | 0..21 | 16 | 2007 | Jaroslaw Wroblewski |
| 22 | 11410337850553 + 475180·19#·n | 0..21 | 15 | 1993 | Paul Pritchard et al. |
| 23 | 20389023122473 + 5785546·23#·n | 0..22 | 17 | 2009 | Eric Markle, PrimeGrid, AP26 |
| 23 | 43760869165417 + 2339805·23#·n | 0..22 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 23 | 56211383760397 + 199678·23#·n | 0..22 | 16 | 2004 | Markus Frind, Paul Jobling, Paul Underwood |
| 24 | 39421708111691 + 9740894·23#·n | 0..23 | 17 | 2009 | Mark Codding, PrimeGrid, AP26 |
| 24 | 167806194923077 + 4935146·23#·n | 0..23 | 17 | 2009 | Andreas Kobara, PrimeGrid, AP26 |
| 24 | 401516288036293 + 608239·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 468395662504823 + 205619·23#·n | 0..23 | 16 | 2007 | Jaroslaw Wroblewski |
| 25 | 2960886048458003 + 2346233·23#·n | 0..24 | 17 | 2009 | BOINC@Poland , PrimeGrid, 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 |
| 26 | 43142746595714191 + 23681770·23#·n | 0..25 | 18 | 2010 | Benoãt Perichon, PrimeGrid, AP26 |
| 27 | 224584605939537911 + 81292139·23#·n | 0..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
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 |
|---|---|---|---|---|---|
| 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 | 11410337850553 + 475180·19#·n (108201410428753) | 0..21 | 15 | 1993 | Paul Pritchard et al. |
| 23 | 403185216600637 + 9523·23#·n (449924511422857) | 0..22 | 15 | 2006 | Markus Frind |
| 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 |
| k | Primes | n's | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 22 | 11410337850553 + 475180·19#·n (108201410428753) | 0..21 | 15 | 1993 | Paul Pritchard et al. |
| 23 | 403185216600637 + 9523·23#·n (449924511422857) | 0..22 | 15 | 2006 | Markus Frind |
| 23 | 100308707032367 + 156722·23#·n (869507044019447) | 0..22 | 15 | 2006 | Markus Frind |
| 23 | 56211383760397 + 199678·23#·n (1036239621869317) | 0..22 | 16 | 2004 | Markus Frind, Paul Jobling, Paul Underwood |
| 24 | 515486946529943 + 136831·23#·n (1217585417914253) | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 468395662504823 + 205619·23#·n (1523454717745013) | 0..23 | 16 | 2007 | 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 |
| 26 | 43142746595714191 + 23681770·23#·n (175223597495211691) | 0..25 | 18 | 2010 | Benoãt Perichon, PrimeGrid, AP26 |
| 27 | 224584605939537911 + 81292139·23#·n (696112717486210091) | 0..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| k | Primes | n's | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 27 | 224584605939537911 + 81292139·23#·n | 0..26 | 18 | 2019 | Rob Gahan, PrimeGrid, AP26 |
| 26 | 89937610947392099 + 78413143·23#·n | 0..25 | 18 | 2018 | Daniel Wimmer, PrimeGrid, AP26 |
| 26 | 271702189272825977 + 67515487·23#·n | 0..25 | 18 | 2018 | Syracuse University, PrimeGrid, AP26 |
| 26 | 55837783597462913 + 62121807·23#·n | 0..25 | 18 | 2018 | Tom Greer, PrimeGrid, AP26 |
| AP25's are not added above here | |||||
| 25 | 188705909233852867 + 44924692·23#·n | 0..24 | 18 | 2017 | Jeff Webster, PrimeGrid, AP26 |
| 25 | 66147818970286411 + 43305462·23#·n | 0..24 | 18 | 2017 | Tim Terry, PrimeGrid, AP26 |
| 25 | 146492329344492673 + 42846350·23#·n | 0..24 | 18 | 2017 | Kyle Schwind, PrimeGrid, AP26 |
| 25 | 293037522812241983 + 42713298·23#·n | 0..24 | 18 | 2017 | Bill Brandt-Gasuen, PrimeGrid, AP26 |
| 25 | 139819001440953661 + 39776662·23#·n | 0..24 | 18 | 2017 | Steven Schapendonk, PrimeGrid, AP26 |
| 25 | 282340243066022987 + 39699195·23#·n | 0..24 | 18 | 2017 | Wolfgang Schwieger, PrimeGrid, AP26 |
| 26 | 48277590120607451 + 37835074·23#·n | 0..25 | 18 | 2017 | Bruce E. Slade, PrimeGrid, AP26 |
| 25 | 96256695044976793 + 37286573·23#·n | 0..24 | 18 | 2017 | Bryan Little, PrimeGrid, AP26 |
| 25 | 320382581839925153 + 36759418·23#·n | 0..24 | 18 | 2017 | David E. Miller, PrimeGrid, AP26 |
| 25 | 172620809233105201 + 35156983·23#·n | 0..24 | 18 | 2017 | Wolfgang Schwieger, PrimeGrid, AP26 |
| 25 | 161146967531777047 + 34071541·23#·n | 0..24 | 18 | 2017 | Gustav Nylinder, PrimeGrid, AP26 |
| 25 | 296950539631234873 + 32252465·23#·n | 0..24 | 18 | 2017 | William Donovan, PrimeGrid, AP26 |
| 25 | 228315362514429847 + 29165712·23#·n | 0..24 | 18 | 2017 | Tuna Ertemalp, PrimeGrid, AP26 |
| 25 | 211810229255864971 + 26698740·23#·n | 0..24 | 18 | 2017 | Erik Klein, PrimeGrid, AP26 |
| 25 | 264703230462051737 + 25208577·23#·n | 0..24 | 18 | 2017 | user Autumn_Beijing, PrimeGrid, AP26 |
| 25 | 73237102080888511 + 23647405·23#·n | 0..24 | 18 | 2017 | Vince Splain, PrimeGrid, AP26 |
| 25 | 233662486570847311 + 23473713·23#·n | 0..24 | 18 | 2017 | Anthony Ayiomamitis, PrimeGrid, AP26 |
| 25 | 92134030929661723 + 23073685·23#·n | 0..24 | 18 | 2017 | Patrik Eriksson, PrimeGrid, AP26 |
| 25 | 234934262624764103 + 22561202·23#·n | 0..24 | 18 | 2017 | Syracuse University, PrimeGrid, AP26 |
| 25 | 171104686521473149 + 19323763·23#·n | 0..24 | 18 | 2017 | Bryan Little, PrimeGrid, AP26 |
| 25 | 295363409604322229 + 17257904·23#·n | 0..24 | 18 | 2016 | Roland Klepel, PrimeGrid, AP26 |
| 25 | 269235074116077859 + 17010331·23#·n | 0..24 | 18 | 2016 | Eirik Lunner, PrimeGrid, AP26 |
| 26 | 142099325379199423 + 16549135·23#·n | 0..25 | 18 | 2016 | Koichi Soraku, PrimeGrid, AP26 |
| 25 | 148375001202532501 + 15439522·23#·n | 0..24 | 18 | 2016 | Eirik Lunner, PrimeGrid, AP26 |
| 25 | 298190132293964681 + 15023287·23#·n | 0..24 | 18 | 2016 | Evelyn Chew, PrimeGrid, AP26 |
| 25 | 117409484129625731 + 15143792·23#·n | 0..24 | 18 | 2016 | user "skgiven", PrimeGrid, AP26 |
| 25 | 240091225937752601 + 14964187·23#·n | 0..24 | 18 | 2016 | Stefan Hofacker, PrimeGrid, AP26 |
| 25 | 228744309021939859 + 14918987·23#·n | 0..24 | 18 | 2016 | David Walling, PrimeGrid, AP26 |
| 25 | 183800317923336901 + 10816541·23#·n | 0..24 | 18 | 2016 | James Krauss, PrimeGrid, AP26 |
| 25 | 290884103695102903 + 12444168·23#·n | 0..24 | 18 | 2016 | Mike Kinney, PrimeGrid, AP26 |
| 25 | 231349139005158193 + 12169920·23#·n | 0..24 | 18 | 2016 | Stefan Larsson, PrimeGrid, AP26 |
| 25 | 203563934890169353 + 11105961·23#·n | 0..24 | 18 | 2016 | Stephen Draycott, PrimeGrid, AP26 |
| 25 | 193513604089287343 + 10189009·23#·n | 0..24 | 18 | 2016 | John S Chambers, PrimeGrid, AP26 |
| 25 | 84314670428700353 + 9954590·23#·n | 0..24 | 18 | 2016 | Ross Goudie, PrimeGrid, AP26 |
| 25 | 327723838632911059 + 10176298·23#·n | 0..24 | 18 | 2016 | Josef Matousek, PrimeGrid, AP26 |
| 25 | 322477370185894411 + 9633039·23#·n | 0..24 | 18 | 2016 | Rafael Trigueiro, PrimeGrid, AP26 |
| 25 | 299460668118437929 + 9026994·23#·n | 0..24 | 18 | 2016 | Hiroyuki Okazaki, PrimeGrid, AP26 |
| 26 | 149836681069944461 + 7725290·23#·n | 0..25 | 18 | 2016 | Takeshi Nakamura, PrimeGrid, AP26 |
| 25 | 124464776168666173 + 2104383·23#·n | 0..24 | 18 | 2016 | Honza Cholt, PrimeGrid, AP26 |
| 25 | 235660731794099011 + 1949500·23#·n | 0..24 | 18 | 2016 | Magnus Karlsson, PrimeGrid, AP26 |
| 25 | 223696034591087459 + 1731112·23#·n | 0..24 | 18 | 2016 | Tom Greer, PrimeGrid, AP26 |
| 25 | 109968592612273693 + 56069187·23#·n | 0..24 | 18 | 2016 | Bryan Little, AP26 |
| 25 | 13124650646226187 + 39527888·23#·n | 0..24 | 18 | 2016 | Bryan Little, AP26 |
| 25 | 38485413788776093 + 34731706·23#·n | 0..24 | 18 | 2016 | Bryan Little, AP26 |
| 25 | 99571126548111473 + 645973·23#·n | 0..24 | 18 | 2016 | Bryan Little, AP26 |
| 26 | 161004359399459161 + 47715109·23#·n | 0..25 | 18 | 2015 | Bryan Little, AP26 |
| 25 | 136314114923771779 + 47064543·23#·n | 0..24 | 18 | 2014 | Bryan Little, AP26 |
| 25 | 18840719948982091 + 49903506·23#·n | 0..24 | 18 | 2014 | Bryan Little, AP26 |
| 26 | 136926916457315893 + 44121555·23#·n | 0..25 | 18 | 2014 | Bryan Little, AP26 |
| 25 | 65921086021334669 + 43905174·23#·n | 0..24 | 18 | 2013 | Bryan Little, AP26 |
| 26 | 3486107472997423 + 1666981·23#·n | 0..25 | 17 | 2012 | James Fry |
| AP24's are not added above here | |||||
| 24 | 35930857390358597 + 24335672·23#·n | 0..23 | 18 | 2010 | Ossi Mauno, PrimeGrid, AP26 |
| 24 | 76189875325979951 + 9176349·23#·n | 0..23 | 18 | 2010 | Oleg Tchij, PrimeGrid, AP26 |
| 26 | 43142746595714191 + 23681770·23#·n | 0..25 | 18 | 2010 | Benoãt Perichon, PrimeGrid, AP26 |
| 25 | 83386545459573043 + 1684949·23#·n | 0..24 | 17 | 2010 | Tigz Mordan, PrimeGrid, AP26 |
| 24 | 78604090316508061 + 8498518·23#·n | 0..23 | 18 | 2010 | Gerrit Slomma, PrimeGrid, AP26 |
| 25 | 18626565939034793 + 30821486·23#·n | 0..24 | 18 | 2010 | Chris Wingate, PrimeGrid, AP26 |
| 24 | 37706304556102387 + 21790711·23#·n | 0..23 | 18 | 2010 | John R. Bailey, PrimeGrid, AP26 |
| 24 | 16509022283270389 + 29463383·23#·n | 0..23 | 18 | 2010 | Gerrit Slomma, PrimeGrid, AP26 |
| 24 | 2687263842007609 + 38710532·23#·n | 0..23 | 18 | 2010 | Richard J. Konish, PrimeGrid, AP26 |
| 24 | 69067110822206929 + 6753086·23#·n | 0..23 | 18 | 2010 | Anonymous, PrimeGrid, AP26 |
| 24 | 35866756599356303 + 19359923·23#·n | 0..23 | 18 | 2010 | Ronald J. Littlefield, PrimeGrid, AP26 |
| 24 | 59742395397388433 + 12519529·23#·n | 0..23 | 18 | 2010 | Alessandro Ceriani, PrimeGrid, AP26 |
| 25 | 25300381597038677 + 28603610·23#·n | 0..24 | 18 | 2010 | Dave Sunderland, PrimeGrid, AP26 |
| 24 | 9029101367197477 + 37882313·23#·n | 0..23 | 18 | 2010 | Roald Solbjørg, PrimeGrid, AP26 |
| 24 | 1580853039814361 + 36803506·23#·n | 0..23 | 18 | 2010 | Anonymous, PrimeGrid, AP26 |
| 24 | 31884137812843999 + 26514806·23#·n | 0..23 | 18 | 2010 | Uros Vizintin, PrimeGrid, AP26 |
| 24 | 73423142215059841 + 4720552·23#·n | 0..23 | 17 | 2010 | Ulrich Hartel, PrimeGrid, AP26 |
| 24 | 57421089862716869 + 12188684·23#·n | 0..23 | 18 | 2010 | Jan Mihrmeister, PrimeGrid, AP26 |
| 24 | 68534294086445963 + 11549562·23#·n | 0..23 | 18 | 2010 | Dave and Antonia Domm, PrimeGrid, AP26 |
| 25 | 42592855872841649 + 19093314·23#·n | 0..24 | 18 | 2010 | Keith Pattenden, PrimeGrid, AP26 |
| 24 | 28806475189976381 + 36028618·23#·n | 0..23 | 18 | 2010 | John Petterson, PrimeGrid, AP26 |
| 25 | 58555890166091939 + 10416756·23#·n | 0..24 | 18 | 2010 | Bryan Little, PrimeGrid, AP26 |
| 24 | 28177336417360261 + 25675304·23#·n | 0..23 | 18 | 2010 | Paweł Kaczmarczyk, PrimeGrid, AP26 |
| 24 | 4076845864885399 + 35779587·23#·n | 0..23 | 18 | 2010 | Edward Gordon, PrimeGrid, AP26 |
| 24 | 24217435709571029 + 24938944·23#·n | 0..23 | 18 | 2010 | Jack England, PrimeGrid, AP26 |
| 24 | 37560025189846517 + 16380129·23#·n | 0..23 | 18 | 2010 | Anonymous, PrimeGrid, AP26 |
| 24 | 22972217245866943 + 23923567·23#·n | 0..23 | 18 | 2010 | Mathias Striecks, PrimeGrid, AP26 |
| 24 | 38156446254970321 + 16171100·23#·n | 0..23 | 18 | 2010 | Anonymous, PrimeGrid, AP26 |
| 24 | 36216875966560057 + 24123429·23#·n | 0..23 | 18 | 2010 | Jon Sonntag, PrimeGrid, AP26 |
| 25 | 49644063847333931 + 7851809·23#·n | 0..24 | 17 | 2010 | Jan Stenzel, PrimeGrid, AP26 |
| 24 | 69844511157803653 + 1408042·23#·n | 0..23 | 17 | 2010 | Geoffrey Leist, PrimeGrid, AP26 |
| 24 | 77964779011754827 + 1320946·23#·n | 0..23 | 17 | 2010 | Bryan Little, PrimeGrid, AP26 |
| 24 | 52284918375969179 + 5444625·23#·n | 0..23 | 17 | 2010 | Charlene Schaar, PrimeGrid, AP26 |
| 24 | 47559343191365491 + 12958707·23#·n | 0..23 | 18 | 2010 | Hubert Stickelberger, PrimeGrid, AP26 |
| 24 | 29578980078344351 + 23462565·23#·n | 0..23 | 18 | 2010 | Donald Hatland, PrimeGrid, AP26 |
| 25 | 46428033558097831 + 12893265·23#·n | 0..24 | 18 | 2010 | Gerrit Slomma, PrimeGrid, AP26 |
| 24 | 22894182607279507 + 19352116·23#·n | 0..23 | 18 | 2010 | Edward Gordon, PrimeGrid, AP26 |
| 24 | 22050598518962261 + 20896917·23#·n | 0..23 | 18 | 2010 | Klaus Bauer, PrimeGrid, AP26 |
| 24 | 36629598861376343 + 11370937·23#·n | 0..23 | 17 | 2010 | Edward Gordon, PrimeGrid, AP26 |
| 25 | 24715375237181843 + 19071018·23#·n | 0..24 | 18 | 2010 | Dan Swearingen, PrimeGrid, AP26 |
| 24 | 35061200218767479 + 10757586·23#·n | 0..23 | 17 | 2010 | Scott Henderson, PrimeGrid, AP26 |
| 24 | 47951620104244397 + 11867770·23#·n | 0..23 | 18 | 2010 | Junyi Guo, PrimeGrid, AP26 |
| 24 | 51098319563712361 + 3993145·23#·n | 0..23 | 17 | 2009 | Uwe Boll, PrimeGrid, AP26 |
| 24 | 29831273365663157 + 16680270·23#·n | 0..23 | 18 | 2009 | Jan Stenzel, PrimeGrid, AP26 |
| 24 | 63445575002484173 + 2307769·23#·n | 0..23 | 17 | 2009 | Robert Scullin, PrimeGrid, AP26 |
| 24 | 18418363373094851 + 16742739·23#·n | 0..23 | 18 | 2009 | Hal Wold, PrimeGrid, AP26 |
| 24 | 49161262681970809 + 10137725·23#·n | 0..23 | 18 | 2009 | Rune Fredriksen, PrimeGrid, AP26 |
| 25 | 29952203513013839 + 8769895·23#·n | 0..24 | 17 | 2009 | David Chevalier, PrimeGrid, AP26 |
| 24 | 61593273431157907 + 446068·23#·n | 0..23 | 17 | 2009 | SETI.USA Cluster, PrimeGrid, AP26 |
| 25 | 45013419998786779 + 9560248·23#·n | 0..24 | 17 | 2009 | Rune Fredriksen, PrimeGrid, AP26 |
| 24 | 4616206285589027 + 27245041·23#·n | 0..23 | 18 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 29051233904541803 + 18765591·23#·n | 0..23 | 18 | 2009 | Krzysztof Koczubik, PrimeGrid, AP26 |
| 24 | 17014421509997489 + 17580742·23#·n | 0..23 | 18 | 2009 | Fred Richard, PrimeGrid, AP26 |
| 24 | 6450318626856161 + 30430175·23#·n | 0..23 | 18 | 2009 | Dave Mumper, PrimeGrid, AP26 |
| 24 | 44265615051136733 + 7821972·23#·n | 0..23 | 17 | 2009 | Jörg Steinmetz, PrimeGrid, AP26 |
| 24 | 3694862982684983 + 28285375·23#·n | 0..23 | 18 | 2009 | Siegfried Niklas, PrimeGrid, AP26 |
| 24 | 22133406276301387 + 15744033·23#·n | 0..23 | 18 | 2009 | Rune Fredriksen, PrimeGrid, AP26 |
| 24 | 7448460750808573 + 27806605·23#·n | 0..23 | 18 | 2009 | Jesper Kågefors, PrimeGrid, AP26 |
| 24 | 28236790404122779 + 13068553·23#·n | 0..23 | 17 | 2009 | Rune Fredriksen, PrimeGrid, AP26 |
| 24 | 20813446029088183 + 11788369·23#·n | 0..23 | 17 | 2009 | Robert Szafrański, PrimeGrid, AP26 |
| 24 | 14698398021483853 + 24362735·23#·n | 0..23 | 18 | 2009 | Pawel Ferus, PrimeGrid, AP26 |
| 25 | 12353443596260323 + 23793841·23#·n | 0..24 | 18 | 2009 | Rune Nordbøe Skillingstad, PrimeGrid, AP26 |
| 24 | 4911007789148401 + 20273384·23#·n | 0..23 | 18 | 2009 | Piotr M. Zalewski, PrimeGrid, AP26 |
| 24 | 48306600448056229 + 1742643·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 36252687055565597 + 2173244·23#·n | 0..23 | 17 | 2009 | Pawel Ferus, PrimeGrid, AP26 |
| 24 | 35973377180183173 + 1622443·23#·n | 0..23 | 17 | 2009 | Jacek Kotnowski, PrimeGrid, AP26 |
| 25 | 46176957093163301 + 1109121·23#·n | 0..24 | 17 | 2009 | Ian Dickinson, PrimeGrid, AP26 |
| 24 | 17868447568925983 + 9564699·23#·n | 0..23 | 17 | 2009 | Senji Yamashita, PrimeGrid, AP26 |
| 24 | 26637789112428883 + 9196753·23#·n | 0..23 | 17 | 2009 | Tim Jungbluth, PrimeGrid, AP26 |
| 24 | 4891686128805269 + 19453568·23#·n | 0..23 | 18 | 2009 | Roald Solbjørg, PrimeGrid, AP26 |
| 24 | 4687877159107031 + 18203167·23#·n | 0..23 | 17 | 2009 | Marcus Rosario, PrimeGrid, AP26 |
| 24 | 1948053460212667 + 17745794·23#·n | 0..23 | 17 | 2009 | Chris Wingate, PrimeGrid, AP26 |
| 24 | 3634080452156039 + 16981607·23#·n | 0..23 | 17 | 2009 | Ian Dickinson, PrimeGrid, AP26 |
| 24 | 28453270488159667 + 6183387·23#·n | 0..23 | 17 | 2009 | Pietari Snow, PrimeGrid, AP26 |
| 24 | 26851278717686693 + 5551454·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 18567190093510343 + 5352686·23#·n | 0..23 | 17 | 2009 | Krzysztof Koczubik, PrimeGrid, AP26 |
| 24 | 4187489431145893 + 15636117·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 3193781476385167 + 15609111·23#·n | 0..23 | 17 | 2009 | John S Morris III, PrimeGrid, AP26 |
| 24 | 24676336785406967 + 5024493·23#·n | 0..23 | 17 | 2009 | Danial L Puckett, PrimeGrid, AP26 |
| 24 | 1220728403569963 + 14348016·23#·n | 0..23 | 17 | 2009 | Martin Siegrist, PrimeGrid, AP26 |
| 24 | 2635631384342027 + 14133381·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 10307159737232191 + 14120563·23#·n | 0..23 | 17 | 2009 | Kevin Erickson, PrimeGrid, AP26 |
| 24 | 18038585883263339 + 3679127·23#·n | 0..23 | 17 | 2009 | Karsten Hilbich, PrimeGrid, AP26 |
| 24 | 28263557466746377 + 3312503·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 25 | 18162964758258289 + 3755664·23#·n | 0..24 | 17 | 2009 | Keith Dale, PrimeGrid, AP26 |
| 25 | 20919497549238289 + 3155495·23#·n | 0..24 | 17 | 2009 | Jochen Beck, PrimeGrid, AP26 |
| 24 | 13678065943093049 + 13223804·23#·n | 0..23 | 17 | 2009 | Bryan Little, PrimeGrid, AP26 |
| 24 | 25545151920212759 + 1140241·23#·n | 0..23 | 17 | 2009 | Paolo Bassi, PrimeGrid, AP26 |
| 24 | 20187352211709911 + 1799216·23#·n | 0..23 | 17 | 2009 | Jeffrey D Sessler, PrimeGrid, AP26 |
| 24 | 20909681071069667 + 234797·23#·n | 0..23 | 17 | 2009 | Tina Kent, PrimeGrid, AP26 |
| 24 | 19471368812966089 + 410682·23#·n | 0..23 | 17 | 2009 | Andreas Mohr, PrimeGrid, AP26 |
| 24 | 10317962076055027 + 10241601·23#·n | 0..23 | 17 | 2009 | Jan-Cornelius Molnar, PrimeGrid, AP26 |
| 24 | 19516186145019209 + 313705·23#·n | 0..23 | 17 | 2009 | Carsten Hartwig, PrimeGrid, AP26 |
| 24 | 7979661543967237 + 9936237·23#·n | 0..23 | 17 | 2009 | Bryan Little, PrimeGrid, AP26 |
| 24 | 39421708111691 + 9740894·23#·n | 0..23 | 17 | 2009 | Mark Codding, PrimeGrid, AP26 |
| 24 | 5531900872160491 + 9383796·23#·n | 0..23 | 17 | 2009 | Paweł Feruś, PrimeGrid, AP26 |
| 24 | 13432401425380607 + 9219580·23#·n | 0..23 | 17 | 2009 | Carsten Hartwig, PrimeGrid, AP26 |
| 24 | 14992521666441877 + 8832442·23#·n | 0..23 | 17 | 2009 | Anonymous, PrimeGrid, AP26 |
| 24 | 167806194923077 + 4935146·23#·n | 0..23 | 17 | 2009 | Andreas Kobara, PrimeGrid, AP26 |
| 25 | 2960886048458003 + 2346233·23#·n | 0..24 | 17 | 2009 | BOINC@Poland, PrimeGrid, AP26 |
| 24 | 7960592659339799 + 2326495·23#·n | 0..23 | 17 | 2009 | Pawel Stachowiak, PrimeGrid, AP26 |
| 24 | 6274259724784693 + 2522655·23#·n | 0..23 | 17 | 2009 | Pawel Stachowiak, PrimeGrid, AP26 |
| 24 | 6872932294461509 + 2042703·23#·n | 0..23 | 17 | 2009 | Krzysztof Koczubik, PrimeGrid, AP26 |
| 24 | 2725131905640097 + 1342336·23#·n | 0..23 | 16 | 2009 | Stefan Smietanowski, PrimeGrid, AP26 |
| 24 | 13785500104035967 + 1004314·23#·n | 0..23 | 17 | 2009 | Krzysztof Koczubik, PrimeGrid, AP26 |
| 25 | 6171054912832631 + 366384·23#·n | 0..24 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 6173646951801709 + 587522·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 3120744164655323 + 851923·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 1630799846252777 + 985151·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 401516288036293 + 608239·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 1368212299836997 + 148682·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 515486946529943 + 136831·23#·n | 0..23 | 16 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 1606021011322579 + 3490622·23#·n | 0..23 | 17 | 2008 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 1564588127269043 + 1249750·23#·n | 0..23 | 16 | 2007 | Raanan Chermoni & Jaroslaw Wroblewski |
| 24 | 468395662504823 + 205619·23#·n | 0..23 | 16 | 2007 | Jaroslaw Wroblewski |
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.
Last updated 29 February 2020.