Introduction
This page records the single largest known case of k simultaneous primes for
each k, and the record history.
The idea is to show the best overall record for different types of
sets with k primes.
p# (called p primorial) is the product of all primes ≤ p, e.g. 10# = 2 · 3 · 5 · 7 = 210.
An expression with p# is often used in prime searches to avoid factors ≤ p.
News
2012
January 24. New record: A second 28-digit 18-tuplet by Raanan Chermoni & Jaroslaw
Wroblewski.
2011
December 31. New record: 29-digit 17-tuplet by Roger Thompson.
December 25. New record: 200700-digit twin prime by Timothy D. Winslow, PrimeGrid, TwinGen, LLR.
December 4. New record: A second 28-digit 17-tuplet by Raanan Chermoni & Jaroslaw
Wroblewski.
December 4. New record: 28-digit 18-tuplet by Raanan Chermoni & Jaroslaw
Wroblewski.
August 25. New record: A second 28-digit 17-tuplet by Roger Thompson.
August 21. New record: 28-digit 17-tuplet by Roger Thompson.
August 9. New record: 28-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
June 20. New record: A seventeenth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
May 28. New record: A third 27-digit 19-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
Also contains record 18-tuplet and 17-tuplet.
May 13. New record: A fifteenth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
April 18. New record: A fourteenth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
April 4. New record: A thirteenth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
March 15. New record: 27-digit 17-tuplet by Roger Thompson.
March 14. New record: A twelfth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
February 28. New record: An eleventh 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
February 19. New record: A tenth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
February 19. New record: A second 27-digit 19-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
Also contains record 18-tuplet.
February 13. New record: 2835-digit CC4 1st kind by Michael Angel, NewPGen,
PrimeForm.
February 11. New record: A ninth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
February 9. New record: 27-digit 19-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
Also contains record 18-tuplet.
February 4. New record: An eighth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
February 4. New record: 27-digit 18-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
January 31. New record: A seventh 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
January 21. New record: A sixth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
January 13. New record: A fifth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
January 6. New record: 1141-digit CC5 2nd kind by Michael Angel, NewPGen,
PrimeForm.
January 2. New record: A fourth 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
2010
December 13. New record: A third 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
November 17. New record: A second 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
November 1. New record: 27-digit 17-tuplet by Raanan Chermoni & Jaroslaw Wroblewski.
August 10. New record: 225-digit CC8 2nd kind by Dirk Augustin, NewPGen,
PrimeForm.
July 28. New record: 1116-digit CC5 2nd kind by David Broadhurst, NewPGen,
PrimeForm.
July 17. New record: 357-digit CC7 1st kind by Dirk Augustin, NewPGen,
PrimeForm.
April 26. New record: A second 27-digit 18-tuplet by Jaroslaw Wroblewski.
March 2. New record: 634-digit CC6 1st kind by Dirk Augustin, NewPGen,
PrimeForm.
January 15. New record: 27-digit 18-tuplet by Jaroslaw Wroblewski.
January 9. New record: 10477-digit CC3 1st kind by Tom Wu, LLR.
2009
August 13. New record: 100355-digit twin prime by (SG Grid), Peter Kaiser, Keith Klahn,
Twin Prime Search, PrimeGrid, NewPGen, tpsieve, LLR.
August 7. New record: 48-digit 13-tuplet by Jens K. Andersen.
July 12. New record: 110-digit CC10 2nd kind by Dirk Augustin, NewPGen,
PrimeForm.
June 22. New record: 301-digit 7-tuplet by Jens K. Andersen.
May 1. New record: 559-digit 6-tuplet by Norman Luhn.
March 24. New record: 1107-digit 5-tuplet by Norman Luhn.
March 11. New record: A fifth 26-digit 18-tuplet by Jaroslaw Wroblewski.
February 20. New record: A fourth 26-digit 18-tuplet by Jaroslaw Wroblewski.
January 26. New record: A third 26-digit 18-tuplet by Jaroslaw Wroblewski.
January 23. This page is moved to http://users.cybercity.dk/~dsl522332/math/simultprime.htm.
January 16. New record: A second 26-digit 18-tuplet by Jaroslaw Wroblewski.
2008
December 30. New record: 64-digit CC12 2nd kind by Dirk Augustin, NewPGen,
PrimeForm.
December 18. New record: 26-digit 18-tuplet by Jaroslaw Wroblewski.
December 14. New record: 25-digit 18-tuplet by Jaroslaw Wroblewski.
December 3. New record: 20-digit AP21 (first entry for 21) by Jaroslaw
Wroblewski.
November 17. New record: 10047-digit triplet by Norman
Luhn, François Morain, NewPGen, LLR, Primo, FastECPP.
October 13. Possible future record: A 10047-digit triplet by Norman Luhn. It contains an
unproven probable prime (proof was announced November 17).
September 16. New record for largest known prime: 12978189-digit Mersenne prime by
Edson Smith, GIMPS.
July 18. New record: A fourth 20-digit AP20 by Jaroslaw Wroblewski. Also contains record
AP19.
July 17. New record: A third 20-digit AP20 by Jaroslaw Wroblewski. Also contains record
AP19.
July 3. New record: A seventh 20-digit AP19 by Jaroslaw Wroblewski.
July 1. New record: 292-digit CC7 1st kind kind by Dirk Augustin, NewPGen, PrimeForm.
June 30. New record: A second 20-digit AP20 by Jaroslaw Wroblewski. Also contains record
AP19.
June 30. New record: A fifth 20-digit AP19 by Jaroslaw Wroblewski.
June 29. New record: A fourth 20-digit AP19 by Jaroslaw Wroblewski.
June 29. New record: A third 20-digit AP19 by Jaroslaw Wroblewski.
June 29. New record: A second 20-digit AP19 by Jaroslaw Wroblewski.
June 28. New record: 20-digit AP20 by Jaroslaw Wroblewski. Also contains record
AP19.
June 28. New record: A fourth 19-digit AP19 by Jaroslaw Wroblewski.
June 27. New record: A third 19-digit AP19 by Jaroslaw Wroblewski.
June 27. New record: 19-digit AP20 by Jaroslaw Wroblewski. Also contains record
AP19.
June 26. New record: 30-digit CC16 2nd kind by Jaroslaw Wroblewski.
June 17. New record: 34-digit CC15 2nd kind by Jaroslaw Wroblewski.
June 12. New record: 28-digit CC16 2nd kind by Jaroslaw Wroblewski.
June 10. New records: 27-digit CC17 2nd kind by Jaroslaw Wroblewski.
Also contains record CC16.
June 8. New record: A second 26-digit CC16 2nd kind by Jaroslaw Wroblewski.
June 5. New records: A second 26-digit CC17 2nd kind by Jaroslaw Wroblewski.
Also contains record CC16.
June 5. New record: 26-digit CC17 2nd kind by Jaroslaw Wroblewski.
May 29. New record: 26-digit CC16 1st kind by Jaroslaw Wroblewski.
April 14. New record: 33-digit 15-tuplet by Jens K. Andersen.
February 29. New record: 218-digit 8-tuplet by Jens K. Andersen.
2007
December 28. New records: 46-digit 14-tuplet by Norman Luhn. Also contains
record 13-tuplet.
March 1. New record: 18-digit AP20 by Jaroslaw Wroblewski.
February 27. New record: 19-digit AP19 by Jaroslaw Wroblewski.
January 15. New record: 58711-digit twin prime by Eric Vautier, Dmitri Gribenko,
Patrick W. McKibbon, Twin Prime Search, PrimeGrid, NewPGen, LLR.
2006
December 30. New record: 17-digit AP20 (first entry for 20) by Jaroslaw
Wroblewski.
December 30. New record: 17-digit AP19 by Jaroslaw Wroblewski.
December 30. New record: 15-digit AP19 (first entry for 19) by Jaroslaw Wroblewski.
September 11. New record for largest known prime: 9808358-digit Mersenne prime by
Curtis Cooper, Steven Boone, GIMPS.
August 7. New record: 7937-digit CC3 1st kind by Michael Paridon, NewPGen,
PrimeForm.
June 30. New record: 475-digit CC6 2nd kind by Dirk Augustin, NewPGen, PrimeForm.
May 3. New record: 51780-digit Sophie Germain prime by Zoltán Járai, Gabor
Farkas,
Timea Csajbok, Janos Kasza, Antal Járai.
April 7. New record: 1069-digit 5-tuplet by Norman Luhn.
February 8. New record: 58-digit CC12 2nd kind by Jens K. Andersen.
January 17. New record: 427-digit 6-tuplet by Norman Luhn.
January 4. New record: 282-digit 7-tuplet by Norman Luhn.
2005
December 25. New record for largest known prime: 9152052-digit Mersenne prime by
Curtis Cooper, Steven Boone, GIMPS.
December 7. New record: 41-digit CC13 1st kind by Dirk Augustin, NewPGen,
PrimeForm.
September 8. New record: 51779-digit twin prime by Zoltán Járai, Gabor Farkas,
Timea Csajbok, Janos Kasza, Antal Járai.
August 7. New record: 187-digit CC9 1st kind by Dirk Augustin, NewPGen,
PrimeForm.
August 6. New record: 206-digit 8-tuplet by Norman Luhn.
June 10. New record: 2354-digit CC4 1st kind by Jiong Sun, NewPGen, PrimeForm.
April 29. New record: 2058-digit quadruplet (4-tuplet) by Norman Luhn, Primo.
April 18. New record: 6038-digit CC3 1st kind by Jiong Sun, NewPGen, PrimeForm.
March 19. New record: 1770-digit CC4 2nd kind by Jiong Sun, NewPGen, PrimeForm.
February 27. New record for largest known prime: 7816230-digit Mersenne prime by
Martin Nowak & GIMPS.
February 7. New record: 33-digit 14-tuplet by Christ van Willegen & Jens K.
Andersen.
February 6. New record: 135-digit 9-tuplet by Norman Luhn.
January 19. New record: 180-digit 8-tuplet by Torbjörn Alm & Jens K. Andersen.
2004
December 1. New record: 32-digit 14- and 15-tuplet by Joerg Waldvogel & Peter
Leikauf.
November 24. New record: 174-digit 8-tuplet by Norman Luhn.
November 7. New record: 418-digit CC6 1st kind by Dirk Augustin, NewPGen,
PrimeForm.
October 24. New record: 5190-digit CC3 2nd kind by Jiong Sun, NewPGen, PrimeForm.
October 19. New record: 26-digit 16-tuplet by Jens K. Andersen.
October 16. This page is published at http://hjem.get2net.dk/jka/math/simultprime.htm.
A year link is to an announcement.
| k | Primes | Type | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 1 | 243112609-1 | Mersenne | 12978189 | 2008 | Edson Smith, GIMPS |
| 2 | 3756801695685 · 2666669 ± 1 | Tuplet (twin) | 200700 | 2011 | Timothy D. Winslow, PrimeGrid, TwinGen, LLR |
| 3 | 914546877 · 234772 · 2n - 1, n=0..2 | CC, 1st kind | 10477 | 2010 | Tom Wu, LLR |
| 4 | 1249097877 · 6599# · 2n - 1, n=0..3 | CC, 1st kind | 2835 | 2011 | Michael Angel, NewPGen, PrimeForm |
| 5 | 80670856865 · 2677# · 2n + 1, n=0..4 | CC, 2nd kind | 1141 | 2011 | Michael Angel, NewPGen, PrimeForm |
| 6 | 37488065464 · 1483# · 2n - 1, n=0..5 | CC, 1st kind | 634 | 2010 | Dirk Augustin, NewPGen, PrimeForm |
| 7 | 162597166369 · 827# · 2n - 1, n=0..6 | CC, 1st kind | 357 | 2010 | Dirk Augustin, NewPGen, PrimeForm |
| 8 | 1148424905221 · 509# · 2n + 1, n=0..7 | CC, 2nd kind | 225 | 2010 | Dirk Augustin, NewPGen, PrimeForm |
| 9 | 65728407627 · 431# · 2n - 1, n=0..8 | CC, 1st kind | 187 | 2005 | Dirk Augustin, NewPGen, PrimeForm |
| 10 | 1070828503293 · 239# · 2n + 1, n=0..9 | CC, 2nd kind | 110 | 2009 | Dirk Augustin, NewPGen, PrimeForm |
| 11 | 24698258 · 239# + 28606476153371 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 104 | 2004 | Norman Luhn & Jens K. Andersen |
| 12 | 13931865163581 · 127# · 2n + 1, n=0..11 | CC, 2nd kind | 64 | 2008 | Dirk Augustin, NewPGen, PrimeForm |
| 13 | 61571 · 107# + 4803194122972361 + 0,2,6,8,12,18,20,26,30,32,36,42,48 | Tuplet | 48 | 2009 | Jens K. Andersen |
| 14 | 381955327397348 · 79# + 18393209 + 0,2,8,14,18,20,24,30,32,38,42,44,48,50 | Tuplet | 46 | 2007 | Norman Luhn |
| 15 | 28320350134887132315879689643840 · 2n + 1, n=0..14 | CC, 2nd kind | 34 | 2008 | Jaroslaw Wroblewski |
| 16 | 2368823992523350998418445521 · 2n + 1, n=0..15 | CC, 2nd kind | 30 | 2008 | Jaroslaw Wroblewski |
| 17 | 11413975438568556104209245223 + 0,4,6, 10,16,18,24,28,30,34,40,46,48,54,58,60,66 | Tuplet | 29 | 2011 | Roger Thompson |
| 18 | 1657958103171451056593631007 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 28 | 2012 | Raanan Chermoni & Jaroslaw Wroblewski |
| 19 | 917810189564189435979968491 + 0,6,10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 20 | 30702013451021547907 + 29# · n, n=0..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 21 | 28112131522731197609 + 19# · n, n=0..20 | AP | 20 | 2008 | Jaroslaw Wroblewski |
Credited programs and projects
Programs:
NewPGen by Paul Jobling.
APSieve
by Michael Bell and Jim Fougeron.
tpsieve
by Geoff Reynolds.
TwinGen by David Underbakke.
PRP by George Woltman.
PrimeForm by the OpenPFGW group with George Woltman.
Proth.exe by Yves Gallot.
LLR by Jean Penné.
Primo (formerly Titanix) by Marcel Martin.
VFYPR by Tony Forbes.
FastECPP (unpublished program) by François
Morain.
The primality proving program is only credited above 300 digits.
Projects:
CP10 by Harvey Dubner, Tony Forbes, Nik Lygeros,
Michel Mizony and Paul Zimmermann.
GIMPS by George Woltman, Scott Kurowski, et al.
Twin Prime Search and PrimeGrid, coordinated by Michael
Kwok, Andrea Pacini, Rytis Slatkevicius.
| k | Primes | Type | Digits | Year | Discoverer |
|---|---|---|---|---|---|
| 1 | 243112609-1 | Mersenne | 12978189 | 2008 | Edson Smith, GIMPS |
| 1 | 232582657-1 | Mersenne | 9808358 | 2006 | Curtis Cooper, Steven Boone, GIMPS |
| 1 | 230402457-1 | Mersenne | 9152052 | 2005 | Curtis Cooper, Steven Boone, GIMPS |
| 1 | 225964951-1 | Mersenne | 7816230 | 2005 | Martin Nowak, GIMPS |
| 1 | 224036583-1 | Mersenne | 7235733 | 2004 | Josh Findley, GIMPS |
| 1 | 220996011-1 | Mersenne | 6320430 | 2003 | Michael Shafer, GIMPS |
| 1 | 213466917-1 | Mersenne | 4053946 | 2001 | Michael Cameron, GIMPS |
| 1 | 26972593-1 | Mersenne | 2098960 | 1999 | Nayan Hajratwala, GIMPS |
| 2 | 3756801695685 · 2666669 ± 1 | Tuplet (twin) | 200700 | 2011 | Timothy D. Winslow, PrimeGrid, TwinGen, LLR |
| 2 | 65516468355 · 2333333 ± 1 | Tuplet (twin) | 100355 | 2009 | (SG Grid), Peter Kaiser, Keith Klahn, Twin Prime Search, PrimeGrid, NewPGen, tpsieve, LLR |
| 2 | 2003663613 · 2195000 ± 1 | Tuplet (twin) | 58711 | 2007 | Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Twin Prime Search, PrimeGrid, NewPGen, LLR |
| 2 | 137211941292195 · 2171960 · n - 1, n=1,2 | CC, 1st kind (Sophie Germain) | 51780 | 2006 | Zoltán Járai, Gabor Farkas, Timea Csajbok, Janos Kasza, Antal Járai |
| 2 | 16869987339975 · 2171960 ± 1 | Tuplet | 51779 | 2005 | Zoltán Járai, Gabor Farkas, Timea Csajbok, Janos Kasza, Antal Járai |
| 2 | 33218925 · 2169690 ± 1 | Tuplet | 51090 | 2002 | Daniel Papp, NewPGen, PRP, Proth.exe |
| 2 | 318032361 · 2107001 ± 1 | Tuplet | 32220 | 2001 | David Underbakke, Phil Carmody, NewPGen, PRP, PrimeForm |
| 2 | 1807318575 · 298305 ± 1 | Tuplet | 29603 | 2001 | David Underbakke, Phil Carmody, NewPGen, PRP, Proth.exe |
| 2 | 665551035 · 280025 ± 1 | Tuplet | 24099 | 2000 | David Underbakke, Phil Carmody, NewPGen, PRP, Proth.exe |
| 3 | 914546877 · 234772 · 2n - 1, n=0..2 | CC, 1st kind | 10477 | 2010 | Tom Wu, LLR |
| 3 | 2072644824759 · 233333 -1, +1, +5 | Tuplet | 10047 | 2008 | Norman Luhn, François Morain, NewPGen, LLR, Primo, FastECPP |
| 3 | 164210699973 · 226326 · 2n - 1, n=0..2 | CC, 1st kind | 7937 | 2006 | Michael Paridon, NewPGen, PrimeForm |
| 3 | 3020255265 · 220023 · 2n - 1, n=0..2 | CC, 1st kind | 6038 | 2005 | Jiong Sun, NewPGen, PrimeForm |
| 3 | 2366867925 · 217206 · 2n + 1, n=0..2 | CC, 2nd kind | 5190 | 2004 | Jiong Sun, NewPGen, PrimeForm |
| 3 | 2288999415 · 215937 · 2n - 1, n=0..2 | CC, 1st kind | 4808 | 2004 | Jiong Sun, NewPGen, PrimeForm |
| 3 | 1110159213 · 215164 · 2n + 1, n=0..2 | CC, 2nd kind | 4575 | 2002 | Michael Angel, Dirk Augustin, Paul Jobling, NewPGen, PRP, Proth.exe |
| 3 | (108748629354 · 4436 · 3251# · (4436·3251#+1) + 210) · (4436·3251#-1) / 35 + 7,11,13 | Tuplet | 4135 | 2002 | David Broadhurst, PrimeForm |
| 3 | (61504372896 · 5119 · 3163# · (5119·3163#+1) + 210) · (5119·3163#-1) / 35 + 5, 7,11 | Tuplet | 4019 | 2002 | David Broadhurst, PrimeForm |
| 4 | 1249097877 · 6599# · 2n - 1, n=0..3 | CC, 1st kind | 2835 | 2011 | Michael Angel, NewPGen, PrimeForm |
| 4 | 119184698 · 5501# · 2n - 1, n=0..3 | CC, 1st kind | 2354 | 2005 | Jiong Sun, NewPGen, PrimeForm |
| 4 | 4104082046 · 4799# + 5651 + 0,2,6,8 | Tuplet | 2058 | 2005 | Norman Luhn, Primo |
| 4 | 1453501013 · 4127# · 2n + 1, n=0..3 | CC, 2nd kind | 1770 | 2005 | Jiong Sun, NewPGen, PrimeForm |
| 4 | 18672891658 · 4099# + 1591788529 + 210n, n=2..5 | CPAP | 1763 | 2003 | Jim Fougeron, Primo |
| 4 | 853877841 · 3571# · 2n ± 1, n=0,1 | BiTwin | 1529 | 2002 | Henri Lifchitz |
| 4 | 1707223185 · 3049# · 2n - 1, n=0..3 | CC, 2nd kind | 1308 | 2002 | Michael Angel, Dirk Augustin, NewPGen, PrimeForm |
| 4 | 356701232 · 3041# · 2n + 1, n=0..3 | CC, 1st kind | 1304 | 2002 | Michael Angel, Dirk Augustin, NewPGen, PrimeForm |
| 4 | 10271674954 · 2999# + 3461 + 0,2,6,8 | Tuplet | 1284 | 2002 | Michael Bell, Michael Davison, Matt Jack, Ronald Lau, Graeme Leese, Ben Lowing |
| 5 | 80670856865 · 2677# · 2n + 1, n=0..4 | CC, 2nd kind | 1141 | 2011 | Michael Angel, NewPGen, PrimeForm |
| 5 | 45008010405 · 2621# · 2n + 1, n=0..4 | CC, 2nd kind | 1116 | 2010 | David Broadhurst, NewPGen, PrimeForm |
| 5 | 424232794973 · 2600# + 43777 + 0,4,6,10,12 | Tuplet | 1107 | 2009 | Norman Luhn, Primo |
| 5 | 283534892623 · 2500# + 1091261 + 0,2,6,8,12 | Tuplet | 1069 | 2006 | Norman Luhn, Primo |
| 5 | 142661157626 · 2411# + 71427757 + 30n, n=0..4 | CPAP | 1038 | 2002 | Jim Fougeron, Primo |
| 5 | 51946113 · 1193# · 2n + 1, n=0..4 | CC, 2nd kind | 509 | 2002 | Dirk Augustin, NewPGen, PrimeForm |
| 5 | 148809453 · 1069# · 2n - 1, n=0..4 | CC, 1st kind | 461 | 2001 | Dirk Augustin, NewPGen, PrimeForm |
| 6 | 37488065464 · 1483# · 2n - 1, n=0..5 | CC, 1st kind | 634 | 2010 | Dirk Augustin, NewPGen, PrimeForm |
| 6 | 601545895935 · 1300# + 43777 + 0,4,6,10,12,16 | Tuplet | 559 | 2009 | Norman Luhn |
| 6 | 37783362904 · 1097# · 2n + 1, n=0..5 | CC, 2nd kind | 475 | 2006 | Dirk Augustin, NewPGen, PrimeForm |
| 6 | 328481121285 · 1000# + 16057 + 0,4,6,10,12,16 | Tuplet | 427 | 2006 | Norman Luhn |
| 6 | 33692909611 · 977# · 2n - 1, n=0..5 | CC, 1st kind | 418 | 2004 | Dirk Augustin, NewPGen, PrimeForm |
| 6 | 44770344615 · 859# + 1204600427 + 30n, n=0..5 | CPAP | 370 | 2003 | Jens K. Andersen, Jim Fougeron, Primo |
| 6 | 2121022995 · 859# + 1204600427 + 30n, n=0..5 | CPAP | 369 | 2003 | Jim Fougeron, VFYPR |
| 6 | 124605675 · 719# · 2n ± 1, n=3..5 | BiTwin | 308 | 2002 | Dirk Augustin |
| 6 | 110282080125 · 700# + 6005887 + 0,4,6,10,12,16 | Tuplet | 301 | 2001 | Norman Luhn, Primo |
| 7 | 162597166369 · 827# · 2n - 1, n=0..6 | CC, 1st kind | 357 | 2010 | Dirk Augustin, NewPGen, PrimeForm |
| 7 | 68841237 · 701# + 2525568581 + 0,2,6,8,12,18,20 | Tuplet | 301 | 2009 | Jens K. Andersen |
| 7 | 1178137903623 · 661# · 2n - 1, n=0..6 | CC, 1st kind | 292 | 2008 | Dirk Augustin, NewPGen, PrimeForm |
| 7 | 251733155478 · 650# + 1146779 + 0,2,8,12,14,18,20 | Tuplet | 282 | 2006 | Norman Luhn |
| 7 | 194688251849 · 601# + x155 + 210n, n=0..6 | CPAP | 259 | 2003 | Jim Fougeron |
| 7 | 7127829789350 · 460# + x177 + 210n, n=0..6 | CPAP | 201 | 2003 | Torbjörn Alm & Jens K. Andersen |
| 7 | 94552004441 · 410# + 165701 + 0,2,6,8,12,18,20 | Tuplet | 178 | 2003 | Jens K. Andersen |
| 7 | 497423806097 · 400# + 380284918609481+ 0,2,6,8,12,18,20 | Tuplet | 173 | 2001 | Norman Luhn |
| 8 | 1148424905221 · 509# · 2n + 1, n=0..7 | CC, 2nd kind | 225 | 2010 | Dirk Augustin, NewPGen, PrimeForm |
| 8 | 330846961 · 503# + 349129635971 + 0,2,6,8,12,18,20,26 | Tuplet | 218 | 2008 | Jens K. Andersen |
| 8 | 12874261020824 · 463# + 88793 + 0,6,8,14,18,20,24,26 | Tuplet | 206 | 2005 | Norman Luhn |
| 8 | 97510235 · 421# + 322355908991 + 0,2,6,8,12,18,20,26 | Tuplet | 180 | 2005 | Torbjörn Alm & Jens K. Andersen |
| 8 | 4319152256906 · 400# + 1277 + 0,2,6,12,14,20,24,26 | Tuplet | 174 | 2004 | Norman Luhn |
| 8 | 65677369861 · 380# + 18000020393471 + 0,2,6,8,12,18,20,26 | Tuplet | 165 | 2003 | Torbjörn Alm & Jens K. Andersen |
| 8 | 15234072433401 · 375# + 43813839521 + 0,2,6,8,12,18,20,26 | Tuplet | 164 | 2002 | Norman Luhn |
| 8 | 243551752728 · 320# + 1277 + 0,2,6,12,14,20,24,26 | Tuplet | 142 | 2001 | Graeme Leese, Michael Bell, Matt Jack, Michael Davison, Ben Lowing, Tim Nightingale, APSieve |
| 9 | 65728407627 · 431# · 2n - 1, n=0..8 | CC, 1st kind | 187 | 2005 | Dirk Augustin, NewPGen, PrimeForm |
| 9 | 90421624808713 · 293# + 103498931 + 0,2,6,8,12,18,20,26,30 | Tuplet | 135 | 2005 | Norman Luhn |
| 9 | 1619062142 · 255# + 53344165991 + 0,2,6,8,12,18,20,26,30 | Tuplet | 111 | 2003 | Thomas Wolter & Jens K. Andersen |
| 9 | 388793398651 · 250# + 1042090781 + 0,2,6,8,12,18,20,26,30 | Tuplet | 110 | 2001 | Norman Luhn |
| 10 | 1070828503293 · 239# · 2n + 1, n=0..9 | CC, 2nd kind | 110 | 2009 | Dirk Augustin, NewPGen, PrimeForm |
| 10 | 24698258 · 239# + 28606476153371 + 0,2,6,8,12,18,20,26,30,32 | Tuplet | 104 | 2004 | Jens K. Andersen |
| 10 | 72613488698235 · 227# + 39058751 + 0,2,6,8,12,18,20,26,30,32 | Tuplet | 103 | 2004 | Norman Luhn |
| 10 | 507618446770482 · 193# + x77 + 210n, n=0..9 | CPAP | 93 | 1998 | Manfred Toplic, CP10 |
| 11 | 24698258 · 239# + 28606476153371 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 104 | 2004 | Norman Luhn & Jens K. Andersen |
| 11 | 35078052 · 157# + 398861548425071 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 70 | 2004 | Jens K. Andersen & Hans Rosenthal |
| 11 | 92119245478633 · 130# + 21816911 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 63 | 2003 | Norman Luhn |
| 11 | 58187756 · 110# + 2320048690691 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 53 | 2003 | Jens K. Andersen |
| 11 | 11450665899501 · 101# + 39058751 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 52 | 2003 | Norman Luhn |
| 11 | 946772106455 · 90# + 872141 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 47 | 2003 | Jens K. Andersen |
| 11 | 3676296453875 · 80# + 10527733922591 + 0,2,6,8,12,18,20,26,30,32,36 | Tuplet | 44 | 2001 | Norman Luhn |
| 12 | 13931865163581 · 127# · 2n + 1, n=0..11 | CC, 2nd kind | 64 | 2008 | Dirk Augustin, NewPGen, PrimeForm |
| 12 | (8428860 · 127# + 212148902055090) · 2n + 1, n=0..11 | CC, 2nd kind | 58 | 2006 | Jens K. Andersen |
| 12 | (4691553 · 97# + 704765232261090) · 2n ± 1, n=0..5 | BiTwin | 44 | 2004 | Jens K. Andersen |
| 12 | 332352838383 · 80# + 38458151 + 0,2,6,8,12,18,20,26,30,32,36,42 | Tuplet | 43 | 2003 | Jens K. Andersen |
| 12 | 5027317106963 · 75# + 1418575498567 + 0,6,10,12,16,22,24,30,34,36,40,42 | Tuplet | 42 | 2001 | Norman Luhn |
| 13 | 61571 · 107# + 4803194122972361 + 0,2,6,8,12,18,20,26,30,32,36,42,48 | Tuplet | 48 | 2009 | Jens K. Andersen |
| 13 | 381955327397348 · 79# + 18393209 + 2,8,14,18,20,24,30,32,38,42,44,48,50 | Tuplet | 46 | 2007 | Norman Luhn |
| 13 | 1753286498051 · 71# · 2n - 1, n=0..12 | CC, 1st kind | 41 | 2005 | Dirk Augustin, NewPGen, PrimeForm |
| 13 | (938719 · 67# + 56461354019070) · 2n+1, n=0..12 | CC, 2nd kind | 33 | 2004 | Jens K. Andersen |
| 13 | 129017282 · 65# + 3001560376931 + 0,2,6,8,12,18,20,26,30,32,36,42,48 | Tuplet | 32 | 2003 | Torbjörn Alm & Jens K. Andersen |
| 13 | 5953176232124 · 50# + 9564671 + 0,2,6,8,12,18,20,26,30,32,36,42,48 | Tuplet | 31 | 2003 | Jens K. Andersen |
| 13 | 1000000008282508019026959814211 + 2,6,8,12,18,20,26,30,32,36,42,48,50 | Tuplet | 31 | 2000 | Joerg Waldvogel & Peter Leikauf |
| 14 | 381955327397348 · 79# + 18393209 + 0,2,8,14,18,20,24,30,32,38,42,44,48,50 | Tuplet | 46 | 2007 | Norman Luhn |
| 14 | 26093748 · 67# + 383123187762431 + 0,2,6,8,12,18,20,26,30,32,36,42,48,50 | Tuplet | 33 | 2005 | Christ van Willegen & Jens K. Andersen |
| 14 | 99999999948164978600250563546400 + 11,13,17,19,23,29,31,37,41,43,47,53,59,61 | Tuplet | 32 | 2004 | Joerg Waldvogel & Peter Leikauf |
| 14 | 1251030012595955901312188450381 + 0,2,6,8,12,18,20,26,30,32,36,42,48,50 | Tuplet | 31 | 2003 | Hans Rosenthal & Jens K. Andersen |
| 14 | 1124495649695212998778414534811 + 0,2,6,8,12,18,20,26,30,32,36,42,48,50 | Tuplet | 31 | 2003 | Thomas Wolter & Jens K. Andersen |
| 14 | 1000000008282508019026959814211 + 0,2,6,8,12,18,20,26,30,32,36,42,48,50 | Tuplet | 31 | 2000 | Joerg Waldvogel & Peter Leikauf |
| 15 | 28320350134887132315879689643840 · 2n + 1, n=0..14 | CC, 2nd kind | 34 | 2008 | Jaroslaw Wroblewski |
| 15 | 107173714602413868775303366934621 + 0,2,6,8,12,18,20,26,30,32,36,42,48,50,56 | Tuplet | 33 | 2008 | Jens K. Andersen |
| 15 | 99999999948164978600250563546400 + 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67 | Tuplet | 32 | 2004 | Joerg Waldvogel & Peter Leikauf |
| 15 | 1251030012595955901312188450381 + 0,2,6,8,12,18,20,26,30,32,36,42,48,50,56 | Tuplet | 31 | 2003 | Hans Rosenthal & Jens K. Andersen |
| 15 | 999999999962618227626700812281 + 0,2,6,8,12,18,20,26,30,32,36,42,48,50,56 | Tuplet | 30 | 2000 | Joerg Waldvogel & Peter Leikauf |
| 16 | 2368823992523350998418445521 · 2n + 1, n=0..15 | CC, 2nd kind | 30 | 2008 | Jaroslaw Wroblewski |
| 16 | 20193491108493165642344880 · 2n + 1, n=0..15 | CC, 2nd kind | 28 | 2008 | Jaroslaw Wroblewski |
| 16 | 1302312696655394336638440 · 2n + 1, n=1..16 | CC, 2nd kind | 27 | 2008 | Jaroslaw Wroblewski |
| 16 | 258296136493222766530020 · 2n + 1, n=0..15 | CC, 2nd kind | 26 | 2008 | Jaroslaw Wroblewski |
| 16 | 127806074555607670094730 · 2n + 1, n=1..16 | CC, 2nd kind | 26 | 2008 | Jaroslaw Wroblewski |
| 16 | 91304653283578934559360 · 2n - 1, n=0..15 | CC, 1st kind | 26 | 2008 | Jaroslaw Wroblewski |
| 16 | 10252256693298561414756287 + 0,2,6, 12,14,20,26,30,32,36,42,44,50,54,56,60 | Tuplet | 26 | 2004 | Jens K. Andersen |
| 16 | 2845372542509911868266807 + 10,12, 16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 25 | 2000 | Joerg Waldvogel & Peter Leikauf |
| 17 | 11413975438568556104209245223 + 0,4,6, 10,16,18,24,28,30,34,40,46,48,54,58,60,66 | Tuplet | 29 | 2011 | Roger Thompson |
| 17 | 1440214698782377829920161551 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 28 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 1193626642220508556578935357 + 0,2,6, 12,14,20,24,26,30,36,42,44,50,54,56,62,66 | Tuplet | 28 | 2011 | Roger Thompson |
| 17 | 1189738328255580204892107257 + 0,2,6, 12,14,20,24,26,30,36,42,44,50,54,56,62,66 | Tuplet | 28 | 2011 | Roger Thompson |
| 17 | 1022964394011851868818264017 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66 | Tuplet | 28 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 947007693039130553371313771 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 917810189564189435979968491 + 10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 865481269763090471673405217 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 783356226566404898275621751 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 772723255295737480928966771 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 766631303403477668452873513 + 0,4,6, 10,16,18,24,28,30,34,40,46,48,54,58,60,66 | Tuplet | 27 | 2011 | Roger Thompson |
| 17 | 730862462126306661846437737 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 711896327852024279416294841 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 663553629195874803746553191 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 646823491570043772214553321 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 635239540644425515747001047 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 597173610793158380251873601 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 572264871047109012711451507 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 528949036434611485053784421 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 506768097227010102028237687 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 446899237331752168867376467 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66 | Tuplet | 27 | 2010 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 388813564304223666130115591 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2010 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 356357564387652563232359861 + 0,6,8, 12,18,20,26,32,36,38,42,48,50,56,60,62,66 | Tuplet | 27 | 2010 | Raanan Chermoni & Jaroslaw Wroblewski |
| 17 | 1302312696655394336638440 · 2n + 1, n=0..16 | CC, 2nd kind | 27 | 2008 | Jaroslaw Wroblewski |
| 17 | 127806074555607670094730 · 2n + 1, n=0..16 | CC, 2nd kind | 26 | 2008 | Jaroslaw Wroblewski |
| 17 | 40244844789379926979140 · 2n + 1, n=0..16 | CC, 2nd kind | 26 | 2008 | Jaroslaw Wroblewski |
| 17 | 2845372542509911868266807 + 4,10,12, 16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 25 | 2000 | Joerg Waldvogel & Peter Leikauf |
| 18 | 1657958103171451056593631007 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 28 | 2012 | Raanan Chermoni & Jaroslaw Wroblewski |
| 18 | 1378931799889412563577465677 + 0,4,10, 12,16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 28 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 18 | 917810189564189435979968491 + 6,10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 18 | 656632460108426841186109951 + 6,10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 18 | 630134041802574490482213901 + 6,10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 18 | 601884606346328759496455407 + 0,4,10,12, 16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 18 | 183837276562811649018077773 + 0,4,6,10, 16,18,24,28,30,34,40,46,48,54,58,60,66,70 | Tuplet | 27 | 2010 | Jaroslaw Wroblewski |
| 18 | 159272023128204753638724847 + 0,4,10,12, 16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 27 | 2010 | Jaroslaw Wroblewski |
| 18 | 51342365971531191697537333 + 0,4,6,10, 16,18,24,28,30,34,40,46,48,54,58,60,66,70 | Tuplet | 26 | 2009 | Jaroslaw Wroblewski |
| 18 | 44453357465442632103684223 + 0,4,6,10, 16,18,24,28,30,34,40,46,48,54,58,60,66,70 | Tuplet | 26 | 2009 | Jaroslaw Wroblewski |
| 18 | 33755528632220819791319947 + 0,4,10,12, 16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 26 | 2009 | Jaroslaw Wroblewski |
| 18 | 23531820918273007548405133 + 0,4,6,10, 16,18,24,28,30,34,40,46,48,54,58,60,66,70 | Tuplet | 26 | 2009 | Jaroslaw Wroblewski |
| 18 | 11298510058634407483251313 + 0,4,6,10, 16,18,24,28,30,34,40,46,48,54,58,60,66,70 | Tuplet | 26 | 2008 | Jaroslaw Wroblewski |
| 18 | 4530085223434265844952597 + 0,4,10,12, 16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 25 | 2008 | Jaroslaw Wroblewski |
| 18 | 2845372542509911868266807 + 0,4,10,12, 16,22,24,30,36,40,42,46,52,54,60,64,66,70 | Tuplet | 25 | 2000 | Joerg Waldvogel & Peter Leikauf |
| 19 | 917810189564189435979968491 + 0,6,10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 19 | 656632460108426841186109951 + 0,6,10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 19 | 630134041802574490482213901 + 0,6,10,16, 18,22,28,30,36,42,46,48,52,58,60,66,70,72,76 | Tuplet | 27 | 2011 | Raanan Chermoni & Jaroslaw Wroblewski |
| 19 | 30702013451021547907 + 29# · n, n=1..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 28195514826922573189 + 29# · n, n=1..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 22855978072012553147 + 29# · n, n=0..18 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 19316011345882272017 + 29# · n, n=1..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 17196286091474921689 + 29# · n, n=0..18 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 14275160661224928539 + 29# · n, n=0..18 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 12167225693277514633 + 29# · n, n=0..18 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 11794878875045456989 + 29# · n, n=0..18 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 11512568504244030089 + 29# · n, n=1..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 19 | 8463643902010505569 + 29# · n, n=0..18 | AP | 19 | 2008 | Jaroslaw Wroblewski |
| 19 | 6619756800771371627 + 29# · n, n=0..18 | AP | 19 | 2008 | Jaroslaw Wroblewski |
| 19 | 2357208768504754823 + 29# · n, n=1..19 | AP | 19 | 2008 | Jaroslaw Wroblewski |
| 19 | 1464072733127278531 + 29# · n, n=0..18 | AP | 19 | 2007 | Jaroslaw Wroblewski |
| 19 | 21167397384723757 + 29# · n, n=0..18 | AP | 17 | 2006 | Jaroslaw Wroblewski |
| 19 | 206063865762371 + 29# · n, n=0..18 | AP | 15 | 2006 | Jaroslaw Wroblewski |
| 20 | 30702013451021547907 + 29# · n, n=0..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 20 | 28195514826922573189 + 29# · n, n=0..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 20 | 19316011345882272017 + 29# · n, n=0..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 20 | 11512568504244030089 + 29# · n, n=0..19 | AP | 20 | 2008 | Jaroslaw Wroblewski |
| 20 | 2357208768504754823 + 29# · n, n=0..19 | AP | 19 | 2008 | Jaroslaw Wroblewski |
| 20 | 843882398917750139 + 29# · n, n=0..19 | AP | 18 | 2007 | Jaroslaw Wroblewski |
| 20 | 19645711679859961 + 29# · n, n=0..19 | AP | 17 | 2006 | Jaroslaw Wroblewski |
| 21 | 28112131522731197609 + 19# · n, n=0..20 | AP | 20 | 2008 | Jaroslaw Wroblewski |
xN is an N-digit number which was used in a record but has no simple expression.
x77 = 545382416838875826681897035
90110659057865934764604873840781923513421103495579
x155 =
573501051193549030504900787301835825160485101511983854806
08646918133804223879167823802443758585361919599047776527963058
419047009660578164772858363185263809
x177 = 248780094097472243404896912 18520204523173891494265205671857771054034765718993
39316926569630394295966974033459801501769756510157 97243607875744813349096932598456481621971387004081
Rules
The listed number of digits is for the median prime, i.e. the i'th of 2i or 2i-1
primes. This prime determines the comparison to other sets.
Using the median is more fair but means the digit count may vary from pages for CC's and
BiTwins.
All primes must be above 1012 (mainly to avoid listing uninteresting k-tuplets
of tiny primes).
All primes must be proven, i.e. not just probable primes.
The record for k=1 is simply the largest known prime.
There are restrictions for k>1 to satisfy a reasonable interpretation of "simultaneous".
Allowed forms include:
Tuplets (primes as closely together as possible).
CC's (Cunningham Chains) of the 1st or 2nd kind.
BiTwins (chains of twin primes).
AP-k (k primes in arithmetic progression): Only allowed with difference k#
or p# or q#, where p and q are the smallest primes above k.
CPAP-k (k consecutive
primes in arithmetic progression): Same restriction as AP-k.
Disallowed forms include:
AP-k and CPAP-k with other differences than explicitly allowed above.
Generalized BiTwins (at bottom of linked page).
Generalized CC's (Cunningham Chains).
I am not aware of a previous comparison of this type and chose rules which seem
fair for the intended purpose. You can suggest rule changes, but arbitrary AP's will never be
allowed.
The rules for allowed forms were made with the idea that k numbers on a specific
predetermined form should be primes simultaneously. The form should usually only
be able to vary one thing, e.g. the first prime.
It should not be possible to compute a large set of primes and then find some subset
of k numbers with something in common, or choose a function which runs through k
already known primes.
An AP search usually first computes a large set of primes and then tests for AP's by
varying both one of the primes and the difference. This is much easier than
finding simultaneous primes. Extreme example: Any 2 numbers are in AP, so the 2
largest known primes will always form an AP-2 which is obviously disallowed (unless it has an allowed difference).
The longest known AP has 25 primes. The largest
known number of simultaneous primes satisfying the rules is an AP21 with
difference 19# and 20 digits. The smallest 18-tuplet above 100 has 25 digits and
may have been harder to find. The smallest AP's above 1000 are usually much smaller
than the smallest case of other forms, because there are more small candidates
without small prime factors. This makes it easier to find small examples of many
simultaneous primes for AP's. Sieving methods means this should not give AP's an
advantage when competing for the largest known case, when other forms have a
known case.
Submissions
I would like to hear of all new records. Please mail any you find or know about,
but note the AP/CPAP rule. Say who should be credited, and which program proved
the primes if they are above 300 digits.
If you are considering a form not mentioned then you can e-mail me and ask
whether it will be allowed. If you think you have found a clever form which
makes a search much easier than the explicitly allowed forms, then the form
almost certainly has a property that will not be allowed. Extreme example: The
n'th prime for k consecutive values of n.
Created and maintained by Jens Kruse Andersen,
jens.k.a@get2net.dk home
Last updated 24 January 2012.