| k | n (first number) | Digits | Origin | Year | Discoverers |
|---|---|---|---|---|---|
| 2 | 282589933 − 1 | 24862048 | Mersenne prime | 2018 | Patrick Laroche, GIMPS |
| 3 | 2618163402417 · 21290001 − 2 | 388342 | Sophie Germain | 2016 | Scott Brown, PrimeGrid, TwinGen, LLR |
| 4 | 1001056355 · 240504 | 12202 | 2017 | Anand Nair, Oscar Östlin, NewPGen, PrimeForm, Prime95, Primo | |
| 5 | 1413937732 · 7937# − 1 | 3414 | 2016 | Oscar Östlin, Primo | |
| 6 | 36133626794 · 6907# · 4 − 4 |
2971 | 2017 | Oscar Östlin, GMP-ECM, PrimeForm, NewPGen, Primo | |
| 7 | 410797845232 · 3499# − 6 | 1493 | 2016 | Oscar Östlin, Primo | |
| 8 | 72·(y−22)·(y−28)·(y−54)·(y−212)/14!−3, where x=429·(250+9096237) and y=(16·x6−72·x4+81·x2−25)2 | 850 | 2009 | David Broadhurst, GMP-ECM, PrimeForm, Pari-GP, Primo | |
| 9 | (y−232)·(y−242)/55440−8, where x=2310·(1030+40790547)+5, t=5·x3−x2−x−1, y=(t·(5·t+9)/2−31)2 | 804 | 2009 | Joe Crump, John Michael Crump GGNFS, GMP-ECM, PrimeForm, Pari-GP | |
| 10 | (2·x3−3·x2−5·x+2)2·(4·x6−12·x5−11·x4+38·x3+13·x2−20·x−6)2/16−8, where x = 1526+10620028 | 552 | 2009 | Joe Crump, John Michael Crump GGNFS, GMP-ECM, PrimeForm, Pari-GP | |
| 11 | (2·x3−3·x2−5·x+2)2·(4·x6−12·x5−11·x4+38·x3+13·x2−20·x−6)2/16−9, where x = 1526+10620028 | 552 | 2009 | Joe Crump, John Michael Crump GGNFS, GMP-ECM, PrimeForm, Pari-GP | |
| 12 | y·(2·y−5)2−9, where x=297+51514439 and y=(x3−13·x−4)2/64 | 521 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 13 | (2·x3−3·x2−5·x+2)2·(4·x6−12·x5−11·x4+38·x3+13·x2−20·x−6)2/16−12, where x = 292+227683166 | 500 | 2009 | Joe Crump, John Michael Crump GGNFS, GMP-ECM, PrimeForm, Pari-GP |
| k | n (first number) | Digits | Origin | Year | Discoverers |
|---|---|---|---|---|---|
| 2 | 277232917 − 1 | 23249425 | Mersenne prime | 2017 | Jonathan Pace, GIMPS |
| 2 | 274207281 − 1 | 22338618 | Mersenne prime | 2016 | Curtis Cooper, GIMPS |
| 2 | 257885161 − 1 | 17425170 | Mersenne prime | 2013 | Curtis Cooper, GIMPS |
| 2 | 243112609 − 1 | 12978189 | Mersenne prime | 2008 | Edson Smith, GIMPS |
| 2 | 232582657−1 | 9808358 | Mersenne prime | 2006 | Curtis Cooper, Steven Boone, GIMPS |
| 3 | 18543637900515 · 2666668 − 2 | 200701 | Sophie Germain | 2012 | Philipp Bliedung, PrimeGrid, TwinGen, LLR |
| 3 | 3756801695685 · 2666669 − 1 | 200700 | twin prime | 2011 | Timothy D. Winslow, PrimeGrid, TwinGen, LLR |
| 3 | 65516468355 · 2333333 − 1 | 100355 | twin prime | 2009 | (SG Grid), Peter Kaiser, Keith Klahn, Twin Prime Search, PrimeGrid, NewPGen, tpsieve, LLR |
| The above was originally record both for proven and prp factors. Later it was only record for proven factors. | |||||
| 3 | 37581121569 · 2333334 − 2 (with a prp factor) | 100355 | Riesel prime (k·2n−1) | 2010 | Jens Kruse Andersen, PrimeForm, based on a Riesel prime found in 2008 by Mark Simpson, NewPGen, PrimeGrid, TPS, LLR |
| The above includes the 100338-digit prp factor (37581121569·2333334−1)/(5·13·73·5743·2342706941). | |||||
| 3 | 2003663613 · 2195000 − 1 | 58711 | twin prime | 2007 | Eric Vautier, Dmitri Gribenko, Patrick W. McKibbon, Twin Prime Search, PrimeGrid, NewPGen, LLR |
| The above was originally record both for proven and prp factors. Later it was only record for proven factors. | |||||
| 3 | 777 · 2247788 − 1 (with a prp factor) | 74595 | Riesel prime (k·2n−1) | 2007 | Jens Kruse Andersen, PrimeForm, based on a Riesel prime found in 2006 by Tony Galvan, NewPGen, Primesearch, LLR |
| The above includes the 74588-digit prp factor (777·2247788+1)/(11·754121). | |||||
| 3 | 1363 · 2246767 − 1 (with a prp factor) | 74288 | Riesel prime (k·2n−1) | 2007 | Jens Kruse Andersen, PrimeForm, based on a Riesel prime found in 2005 by Peter Benson, NewPGen, LLR |
| The above includes the 74281-digit prp factor (1363·2246767+1)/(3·5·7·132·743). | |||||
| 3 | 2347 · 2223281−1 (with a prp factor) | 67218 | Riesel prime | 2007 | Jens Kruse Andersen, PrimeForm, based on a Riesel prime found in 2005 by Thomas Ritschel, NewPGen, LLR |
| The above includes the 67208-digit prp factor (2347·2223281+1)/(3·5·197·1213·1783). | |||||
| 3 | 3045 · 2215472 − 2 (with a prp factor) | 64868 | Riesel prime | 2007 | Jens Kruse Andersen, PrimeForm, based on a Riesel prime found in 2005 by Chris Chatfield, NewPGen, LLR |
| The above includes the 64853-digit prp factor (3045·2215472−2)/(2·17·1151·27067·173473). | |||||
| 3 | 38602791 · 2200026 − 2 (with a prp factor) | 60222 | Riesel prime | 2007 | Jens Kruse Andersen, PrimeForm, based on a Riesel prime from an unsuccessful twin prime search in 2004 by Jiong Sun, NewPGen, LLR |
| The above includes the 60203-digit prp factor (38602791·2200026−1)/(577849·2645749·3427009). | |||||
| 4 | 1001056355 · 240504 (with a prp factor) | 12202 | prp: 2015 proven: 2017 | Oscar Östlin, NewPGen, PrimeForm, Prime95 | |
| The above originally included the 12174-digit prp factor (1001056355·240504+1)/(3·11·17·9743·2723504659773219359123). The prp was proved by Anand Nair in 2017 and it became the record for k=4 with proven prime factors. |
|||||
| 4 | 245363571 · 235426 − 3 | 10673 | Sophie Germain | prp: 2010 proven: 2012 | Tom Wu, GMP-ECM, PrimeForm, Primo, based on a Sophie Germain prime from a CC3 search using LLR |
| The above originally included the 10630-digit prp
factor (245363571·235426-3)/(3·349·111602773267·953666301013·43440278284896679). The prp factor was later proved and the above became the record for k=4 with proven prime factors. |
|||||
| 4 | 378149751 · 227186−2 | 8193 | Sophie Germain | 2010 | Tom Wu, Geoffrey Hird, PrimeForm, Primo, LLR, based on a Sophie Germain prime from a CC3 search |
| 4 | 25390425 · 218703−1 | 5638 | CC2 (2nd kind) | 2010 | Tom Wu, PrimeForm, Primo, LLR, based on a CC2 (2nd kind) from a CC3 search |
| 4 | 21996007 · 233337 (with a prp factor) | 10043 | AP4 search | 2007 | Jens Kruse Andersen, PrimeForm, based on a prime from an AP4 search in 2007 by Ken Davis, NewPGen, PrimeForm |
| The above includes the 10039-digit prp factor (21996007·233337+1)/(3·5·1129). | |||||
| 4 | 2989530439 · 14489#/5 − 1 (with a prp factor) | 6223 | twin prime | 2007 | Christophe Clavier, GMP-ECM, PrimeForm, based on a twin prime found by Norman Luhn, APSieve, PrimeForm |
| The above includes the 6203-digit prp factor (2989530439·14489#/5+2)/(23·943127·5020192965913). | |||||
| 4 | 297079965 · 217434−1 | 5257 | CC2 (2nd kind) | 2009 | Tom Wu, PrimeForm, Primo, LLR, based on a CC2 (2nd kind) from a CC3 search |
| 4 | 240819405 · 213879 | 4187 | CC2 (2nd kind) | 2008 | Matthew Peets, Jens Kruse Andersen, Primo, PrimeForm, based on a CC2 (2nd kind) in 2000 by Markus Frind, Proth.exe |
| 4 | 240819405 · 213879 (with a prp factor) | 4187 | CC2 (2nd kind) | 2007 | Jens Kruse Andersen, PrimeForm, based on a CC2 (2nd kind) in 2000 by Markus Frind, Proth.exe |
| The above is a former record for prp factors allowed at a time
when it included the 4178-digit prp factor (240819405 · 213879+3)/(3·13·43·358877).
This factor was proved prime in 2008 by Matthew Peets with Primo, making it the record
at the time for k=4 with proven
prime factors. During the time it was record for k=4 with prp's allowed, the proven records for k=4 were part of k=5 records with 2063 and later 2135 digits. |
|||||
| 4 | 136857 · 211608 − 1 | 3500 | twin prime | 2007 | Donovan Johnson, GMP-ECM, Primo, based on a twin prime found by Gary Barnes, NewPGen, LLR |
| 4 | 447295839 · 27061 − 3 | 2135 | 2007 | Donovan Johnson, NewPGen, LLR, GMP-ECM, Primo | |
| 5 | 7096755082 · 5021# − 2 | 2158 | BiTwin | 2015 | Oscar Östlin, NewPGen, PrimeForm |
| 5 | 4790484140 · 5021# − 2 | 2158 | BiTwin | 2015 | Oscar Östlin, NewPGen, PrimeForm |
| 5 | 14635080068 · 5011# − 2 | 2155 | BiTwin | 2012 | Dirk Augustin, NewPGen, PrimeForm |
| 5 | (y−114)·(y−352)·(y−472)·(y−942)·(y−1462)·(y−1482)/m−2, where b=5026578700, m=13416739015680·b, x=(267+28683395)·b, y=(8·x6+24·x5+50·x4+54·x3+41·x2+12·x−148)2 | 2139 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 5 | 447295839 · 27061 − 3 | 2135 | 2007 | Donovan Johnson, NewPGen, LLR, GMP-ECM, Primo | |
| 5 | 1749900015 · 26820 − 4 | 2063 | CC3 | 2005 | Jens Kruse Andersen, PrimeForm, Primo, based on a CC3 in 2001 by Paul Jobling, Dirk Augustin, NewPGen, Proth.exe |
| 6 | (y−114)·(y−352)·(y−472)·(y−942)·(y−1462)·(y−1482)/m−3, where b=5026578700, m=13416739015680·b, x=(267+28683395)·b, y=(8·x6+24·x5+50·x4+54·x3+41·x2+12·x−148)2 | 2139 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 6 | (y−114)·(y−352)·(y−472)·(y−942)·(y−1462)·(y−1482)/m−4, where b=4011209802600, m=16812956160·b, x=5000001251617·b and y=(2·x6+3·x3−148)2 | 1803 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 6 | (y−114)·(y−352)·(y−472)·(y−942)·(y−1462)·(y−1482)/m−3, where m=67440294559676054016000 and y=(m·(1096+10624986)+22)2 | 1404 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 6 | (y−222)·(y−612)·(y−862)·(y−1272)·(y−1402)·(y−1512)/m−2, where m=67440294559676054016000 and y=(m·(1096+9581328)+22)2 | 1404 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 6 | (y−114)·(y−352)·(y−472)·(y−942)·(y−1462)·(y−1482)/m−5, where m=67440294559676054016000 and y=(m·(1071+145589)+22)2 | 1104 | 2007 | David Broadhurst, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 6 | 21247003564 · 2411# | 1037 | AP8 search | 2007 | Jens Kruse Andersen, PrimeForm, Primo, based on a prime from an AP8 search in 2003 by Paul Underwood, Markus Frind, NewPGen, PrimeForm |
| 7 | (y−114)·(y−352)·(y−472)·(y−942)·(y−1462)·(y−1482)/m−4, where m=67440294559676054016000 and y=(m·(1096+10624986)+22)2 | 1404 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 7 | 21247003564 · 2411# − 1 | 1037 | AP8 search | 2007 | Jens Kruse Andersen, PrimeForm, Primo, based on a prime from an AP8 search in 2003 by Paul Underwood, Markus Frind, NewPGen, PrimeForm |
| 8 | (y−232)·(y−242)/55440−7, where x=2310·(1030+40790547)+5, t=5·x3−x2−x−1, y=(t·(5·t+9)/2−31)2 | 804 | 2009 | Joe Crump, John Michael Crump GGNFS, GMP-ECM, PrimeForm, Pari-GP | |
| 8 | (y−232)·(y−242)/55440−6, where x=(887040·(9012+111833012))3 and y=(x·(5·x+9)/2−31)2 | 703 | 2009 | Joe Crump, John Michael Crump GGNFS, GMP-ECM, PrimeForm, Pari-GP | |
| 8 | (y−232)·(y−242)/55440−6, where x=(1320·(1022+1932187))3 and y=(x·(5·x+9)/2−31)2 | 600 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 8 | y·(y−23)·(y−41)·(y−64)/55440−2, where y=(13860·(1060+1898683))2 | 509 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 8 | y·(y−23)·(y−41)·(y−64)/55440, where y=(13860·(1060+720251))2 | 509 | 2007 | David Broadhurst, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 9 | (y−34)·(y−210)/55440, where x=(1320·(5·1023+7574922))3 and y=(x·(5·x+9)/2−31)2 | 641 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 9 | (y−232)·(y−242)/55440−7, where x=(1320·(1020+13065906))3 and y=(x·(5·x+9)/2−31)2 | 552 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 9 | 4·(x−1)2·(x2+x+1)2·(8·x6−16·x3+3)2−8, where x=2·1028+2204662 | 512 | 2009 | Joe Crump, John Michael Crump, GMP-ECM, GGNFS, Pari-GP | |
| 9 | y·(y−23)·(y−41)·(y−64)/55440−3, where y=(13860·(1060+1898683))2 | 509 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 10 | y·(2·y−5)2−7, where x=297+51514439 and y=(x3−13·x−4)2/64 | 521 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 10 | y·(2·y−11)2/9−9, where x=3·1028+45140566 and y=(2·x3−10·x−3)2 | 515 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 10 | 4·(x−1)2·(x2+x+1)2·(8·x6−16·x3+3)2−9, where x=2·1028+2204662 | 512 | 2009 | Joe Crump, John Michael Crump, GMP-ECM, GGNFS, Pari-GP | |
| 10 | y·(y−23)·(y−41)·(y−64)/55440−4, where y=(13860·(1060+1898683))2 | 509 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 11 | y·(2·y−5)2−8, where x=297+51514439 and y=(x3−13·x−4)2/64 | 521 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 11 | y·(2·y−11)2/9−10, where x=3·1028+45140566 and y=(2·x3−10·x−3)2 | 515 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| k | n (first number) | Digits | Origin | Year | Discoverers |
|---|---|---|---|---|---|
| 13 | (2·x3−3·x2−5·x+2)2·(4·x6−12·x5−11·x4+38·x3+13·x2−20·x−6)2/16−12, where x = 292+227683166 | 500 | 2009 | Joe Crump, John Michael Crump GGNFS, GMP-ECM, PrimeForm, Pari-GP | |
| 12 | y·(2·y−5)2−9, where x=297+51514439 and y=(x3−13·x−4)2/64 | 521 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 11 | y·(2·y−11)2/9−10, where x=3·1028+45140566 and y=(2·x3−10·x−3)2 | 515 | 2009 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP | |
| 10 | y·(y−23)·(y−41)·(y−64)/55440−4, where y=(13860·(1060+1898683))2 | 509 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 9 | Same as for k=10 | 509 | 2007 | David Broadhurst, GGNFS, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 8 | y·(y−23)·(y−41)·(y−64)/55440, where y=(13860·(1060+720251))2 | 509 | 2007 | David Broadhurst, GMP-ECM, Msieve, PrimeForm, Pari-GP, Primo | |
| 7 | 21247003564 · 2411# −1 | 1037 | AP8 search | 2007 | Jens Kruse Andersen, PrimeForm, Primo, based on a prime from an AP8 search in 2003 by Paul Underwood, Markus Frind, NewPGen, PrimeForm |
| 6 | Same as for k=7 | 1037 | AP8 search | 2007 | Jens Kruse Andersen, PrimeForm, Primo, based on a prime from an AP8 search in 2003 by Paul Underwood, Markus Frind, NewPGen, PrimeForm |
k=2: 24862048 digits.
n = 2^82589933-1 (Mersenne prime)
n+1 = 2^82589933
k=3: 388342 digits.
(n = 2618163402417*2^1290001-2)
n = 2*(2618163402417*2^1290000-1) (2*prime)
n+1 = 2618163402417*2^1290001-1 (prime)
n+2 = 2^1290001*3^2*290907044713
k=4: 12202 digits.
(n = 1001056355*2^40504)
n = 2^40504*5*13*15400867
n+1 = 3*11*17*9743*2723504659773219359123*p12174
n+2 = 2*(1001056355*2^40503+1) (2*prime)
n+3 = 1001056355*2^40504+3 (prime)
Primo certificate for p12174 is in the Download link under "Primality" at
http://www.factordb.com/index.php?id=1100000000777825386.
k=5: 3414 digits.
(n = 1413937732*7937#-1)
n = 579065494045723757809670167419583*p3381
n+1 = 2^2*397*521*1709*7937#
n+2 = 1413937732*7937#+1 (prime)
n+3 = 2*(706968866*7937#+1) (2*prime)
n+4 = 3*(1413937732*7937#/3+1) (3*prime)
Primo certificate for p3381 is in
http://primerecords.dk/certif/k5dig3414.zip.
k=6: 2971 digits.
(n = 36133626794*6907#*4-4)
n = 2*2*p2970
n+1 = 3*82988833*p2963
n+2 = 2*p2971
n+3 = 36133626794*6907#*4-1 (twin prime)
n+4 = 2^3*18066813397*6907#
n+5 = 36133626794*6907#*4+1 (twin prime)
n-1 = 5*41479*9482120399*692307937*131322060679698527*181584030482638864427*91941573397640066926253*590631941644223822689303*c2863
n+6 = 2*29202485169059*8174320284005081474951269*c2932
Primo certificate for p2963 is in the Download link under "Primality" at
http://factordb.com/index.php?id=1100000000935801201.
k=7: 1493 digits.
(n = 410797845232*3499#-6)
n = 2*3^2*173273*1166749799457452217869285429*p1459
n+1 = 5*2385113*p1486
n+2 = 2^2*(102699461308*3499#-1) (2^2*prime)
n+3 = 3*12418053157*305955714712481*893413432297756873*p1450
n+4 = 2*(205398922616*3499#-1) (2*prime)
n+5 = 410797845232*3499#-1 (prime)
n+6 = 2^4*3499#*25674865327
Primo certificates for the non-trivial proofs p1459, p1486 and p1450 are in
http://primerecords.dk/certif/k7dig1493.zip.
k=8: 850 digits.
(n = 72*(y-2^2)*(y-2^8)*(y-5^4)*(y-21^2)/14!-3,
where x=429*(2^50+9096237) and y=(16*x^6-72*x^4+81*x^2-25)^2)
n = 3*p849
n+1 = 2*9277*9862129*3520556383*110169881898372133*p812
n+2 = p850a
n+3 = 2^14*3^12*5^2*11*13*17^2*71*97*127*151*163^2*191^2*463*857*911*2039^2*3121^2*3671^2*27617*94559*105397*334363*818093*1081361*1094881*8032223*8673499*28379557*379425797*1928145619*33487856029*75107484533*1932425260981*9461343831419^2*14124202973927*43900119421747^2*78263386849913*87716528455057^2*138911112511313*332653160927821*4685494971602672527*22578112014348327386927*122263255338736449624461*53567894301100706956015487*55113771707297541288419321*255922537221444436661056643^2*145432951087573335616229844260519*4900032397637694849377097976316041*926379724905047662387400079983152469*3535157730088988059299768749292371183*55948241477020996875929924604390118463*3743940973665875353073451608837014391941255351681351250046220217081500959197080629383389*203171345168977554057804535252209695979589563860186352662241349515657691011371586515901396031719887008989179
n+4 = 7*23^3*37^2*79*83*131*167*257*359*1399*2269*3931*5113*7523*13103*89101*132299*213281*263089*851549*4249391*6665843*111758329*539610871*752205743*1174151543*31955728957*36525034891*69815079791*268077520427*1691563012343*2886922982807*4975707238891*59360469637561*212588265532079*5113751487906046691*8073995731340531749*85132972807816646399*698440825237695963167*235466779559351867156089*8638974032935885408356031*20630841274038888637908941*23544276279027077345208343691*65273164625411946236961729023*2805682321785697112266888628881*7463890134212550090674435831341*301121597780546403842070623578537*5865153635770685042150185867353407929*6528501668900596816131526302366696161*297583971826860297080110478721751715863*609922395430796703126238760038151517372382559*11490578631615430896997480933928606585904494641396939747239142030295759*544384709264090687490313199283910398850735650743667894915780898418916503024247696440699877047
n+5 = 2*p850b
n+6 = 3*19*1009*p845
n+7 = 2^2*208577*p844
p850a and p850b are different 850-digit primes.
n-1 = 2^2*95819*27609121*c837
n+8 = 5*311137*114685080442874011127*c824
n-1 and n+8 are unlikely to have other prime factors below 10^25.
There is a checked verification for k=8 in
http://physics.open.ac.uk/~dbroadhu/cert/ifacgast.zip.
k=9: 804 digits.
(n = (y-23^2)*(y-24^2)/55440-8, where x=2310*(10^30+40790547)+5, t=5*x^3-x^2-x-1, y=(t*(5*t+9)/2-31)^2)
n = 2^2*4261*75337*110917*574283*15755933*255946099*166880080123*p757
n+1 = 3^3*5*331*p799
n+2 = 2*61*2011*10601*444546250219*p783
n+3 = 2423*101557755913*266403633889*p778
n+4 = 2^11*3*13*19*23^2*43*47*53*71*79*103*113*131*313*389*619*1093*1579*7451*13309*33569*76757*92627*1319261*1114238843*18290398183*489625035961*1270641752531*52875541480393*3354903562911899*52004118381270427*63033107717874313*717795020292627461*23659571581081290354933869*3702831616477791607996624583*257083647123710228163986163068933*72995580604330472767675305956448577*5650656063178953440230244461478711321*103254525137730884356325462161948293611*32639399163965702665322588630095762611168465302871817*290963797629785789829587858554435806099102183324391210019*344388126566808701830834392491420283719805371748824281502279*25680373978698894578006006245685748564269008939777185093119975580853*7423574759509261460819979768473950225646324801051680510010128954872507181358849536795703*15516753433189944918910992099992845607888494478960888121798680997688704717693550507172185714219
n+5 = 7*17*37*157*283*317*419*691*1433*1901*1913*6689*2439953*2653993787*3079307041*6361306879*1848987653587*147279592197387043*1289593158350311282393141*258152730541656904182113569565579770280957857*1890907634926683857961173716314441547212123672350629*1159459696029019298615512562569201052337715269676992150571*101395877354331666235965890216235926227140750818573375760797579*4799995106140499339265636929746263616111770974981245944687585577*61795841127498738668772363572553955238962417043039906825237172011091635954239597*5715444996056603421288522870733528787435898718545067670986221446315823787588705707749193186082986329937601489837900897917020845211645033463227*54076763599622413525359748434676447007280788635568722994736593287262429063324143776398234293027462569762217074967207645461719612421989698929945491374185704164228447081879263525107032319
n+6 = 2*5*p803a
n+7 = 3*p803b
n+8 = 2^2*11^2*29*73*643*727*3547*86851*915283*4719726079*14266216717*19750153781*172788128959*17346192250609*365172908112440801*11349953878687101971*2561508111902244874441*79444083751332871206115014253*450306550911446554674008453280965308352893868510532064297*263835423801369863013730916110752495832191782138880338638851594918414481883131465152263052713273663*61631955000000000000007542003471783026400000307642447106035732572941622967899510259568649113820727673*3649682510725537007425276195322911915332208177782676302046766103340728769618932807593262154741181712973564003103*329717977532262240764745389133388989475063541661399274000306678512865674777976547994209353911413438933330737758598303758122047*36666734407117219300514766129735427723031863623789410226119966507111395237732790320214641435454433341255486268836866249362203194571981532931307
p803a and p803b are different 803-digit primes.
n-1 = 41*535101467*c793
n+9 = 599*919*3779*42649*16228241*61486981526111*c769
~2000 ECM curves with B1=1000000 and ~500 with B1=3000000
There is a checked verification for k=9 in
http://immortaltheory.com/cnt/VERIFY_804.gp.
k=10: 552 digits. Same as n+1 to n+10 in record for k=11.
k=11: 552 digits.
(n = (2*x^3-3*x^2-5*x+2)^2*(4*x^6-12*x^5-11*x^4+38*x^3+13*x^2-20*x-6)^2/16-9, where x=15^26+10620028)
n = 3^3*7*11*29*797*1039*1103*1427*3251*3259*39103*204013*535169*272886083*4121448017*21598899661*409072561753*519537015073*858143907144487*2678465528135027*692182816348603133*439911034040804233169*153715102416475789067816753*54797332294839573582156690577*440167453052484845395296777659759700025866945940187*1490753961804495511472010679553553719994851286296636790921170578750322611815602909990587172619578278369031028762662879076160944273007*1450680572742737082125066423130337717154786020427056296586957684578732972366536675987593477013262565042583501426333299494101797785885285518865883802022347
n+1 = 2^3*43487*34434429143*2671318275315433*144391571132721412007659975773678785330610089905539137*940513682365964521411580467479612497184247667346713718331165504974197162360066016886090000157831159486454280417507615487658913*23697051431951006121549913068636179160276931840635795389569544502842632838052313094193084394909824275746194884055203476348365691538147573536676825410145438263097877759897321813731363102528352135040888060614361490454942144750332454109731385394101534920099785563323780503177717538198321927198981106741255758046649749837975685343085297371886447
n+2 = 55337*p547
n+3 = 2*3*5*47*331*359*1291*986981*2002953749*12712144849384397587542162535943365250597239073952624539389402782482235601428590827634430510713300948536358810305637766909036928823755797859532604601090124387614285823*18945361041423043311319369017577973246669307283626167890232229413293367285951608091227742359201034685435855643761232140422680237863456889876666739738438100498603887595881151447426043074385956576132662399924073599216085551343603013723886900327594151459667641630117816913419048703647696736246088622541342174916158369771383055067726471952087324924256509511261829
n+4 = 31*89*181*6361*p542
n+5 = 2^2*17*1613*3607*69127*424841*425109049*6325447759*10312284591953317*746927965061759029*19767145144099170403669*123791177180274476752043071436801231*2952042386335628044687126353949453187623*178481957758622318842860130819690665371964998256863*300927303212925356381171755013770189743897160452800095868728344049290879607763303843462053162212559*1102981670154685461815124635819650935110418538347527302950473117871569767406358091115131099626671177611316414412260909513756090628405728547752055120012571638940984062942925356854780462161945601138704897404375003740810066511414451418641
n+6 = 3*457*6781*13042499*215026223*2776982627*945861378949*25483746753600361*1225228327490491722399383*3048651749705940194612034637*98314563435243698633253261744130446808398230218091*1246623054122463570492075535372545413205158962595747661882605745363469309593103195271046193*1581035282528170936207659753899395448814090750468632122640146026524635806886917078047742443962934242556072610589811768073747816162973*81527183254529280633866482960793949729652462638349992191744411300813558915055882521129101344915359014881870035032987964636684423048773673044373977789429425279536633077
n+7 = 2*7*199*383*22769*1981631*228901271*7062362423*9533727384721*906067834230707143*64730151242356234799*107292386616952796862417353040542655607*991359526871128739001255602307265276247*833250990595522848394855160380170407624056328390633*118542861243684331346133060578940628788108164945643764750449504847*313137269528035784179493718338224423165522666921145657071280783527794991*53128518216684001749484129438868239369239300198595888609347503993013015448440073685348575297*13538811187409942704866045458529013717912051438213362623493323310488245254431519469209178056918451750559268479
n+8 = 5^2*13*109*157*179*10354103*461033426457284890164569*1074664477013774541471015856723*236379140653122682533795791912035915737436342452064899707*836594247678316140409926696591950566029481721475866669170339474308266771*19718256313056761286183243497093070095538402382589116249875457477348917609808894542321259874568820933871344039*5171259921575516811118212485227773865963786606652017136230584806842149049773673503246391413166744435127670198554010570357245742073008066896203908314286152240747748816381591037006910281196887913693451767471640477347962041531413024457148719868747
n+9 = 2^8*3^2*41^2*103^2*27982299736623709616597^2*7594909640655948575910427^2*17674809061335434031710009^2*110187914144408232433740027965203^2*4341485828238153721926363725581225345540759229754953224439159767^2*881026963019664248702772688343335649876074773037498980427802485251903777999983143151021705494380819239^2
n+10= 1061*1877*p545
n-1 = 2*14723*c547
n+11= 2*11^2*19*1884523*5051815528104032427515047480883*19320078030071947802324076404344843659439811*c468
~2000 ECM curves with B1=1000000
There is a checked verification for k=11 in
http://immortaltheory.com/cnt/verify_k11_c552.gp.
k=12: 521 digits.
(n = y*(2*y-5)^2-9, where y=(x^3-13*x-4)^2/64 and x=2^97+51514439)
n = 2^5*3^2*7*13*17*887*2437*13339*30089*49787*295847*699089*631106473*11758433179*73686673513*7997685879317*360477880611768553*518091460634382709*15050964904693165182299*17917308789525613965551*1171398140920239420717280663859*5728960191355159203801240850367037084832744087*92392243208124716402416696390826938854637442681347332860841289*121236226055886382351608060819534752703322469026614011168245516475418263183019798929549185071705234086761*918349743607285428786924077404455720960195812218071791254827049991798749780311888599251213291805037642077080458866120179
n+1 = 257*23296519*1242096601*3048355236167*67973327977858879*135996879951639809*606635802341662435132175179349293714549272051430455678036875643598915349310782211721973275723042298531808064519*476049241189471357760864128686221766350096529085909228324245109554467660977093847926313967394428260917981014913568279257549965254742677328761804745011575095142481178743211265579508624993461874944839192347838538543548727181449071950041770613073355148416139112738412832441110967186585544024820400112942605725121208812404616109968539147733573432097
n+2 = 2*6857*p517
n+3 = 3*5*43*863*1747*3581*21019*179951*2665988320602598119712037148890597589508913576108560151252151200435666518468042695203986763274362208546291758588761326187257881915092235910607165312317946140409466980679*1723456019694535985024052831278769982409850693731036183019401249854005649027058715170650019790949774813382424909094653917094387303200560837954607615294370400024910950769859150808458713206581975301028261726486069902159812186060487965885791282409908342513319078236674752345844445787295726398459700590072492402991653840119101403324069
n+4 = 2^2*59*22091*35999*p510
n+5 = 11*29*101*571*3691*5419*65203*1295389*2920079*138203671*6079815041*6807805279087*4214598470855783*252775275481837740704983146954700963713586730153850922755617*576545967736398377880005536912745355844295932795061127507304915071378800145932900134220140776910446588573801936420449624688244935867703659255654133*189791581562086004182786053911463893927100346909312552275180515756725069187771439438525215323275001305235375297858441337994282392220999296010244121821682564249311868864678056128116961654457526895927237549982385731864650276028768837996759
n+6 = 2*3*54881331191*3549757051333*112712844601859*345742433728700149*61347189073381493207918536815318678415683328792843643*14608757911619482325513417398922661995740823683847143154630043302377343949721739*2384710061042407663350652645312256059993060891913655975778396102465978591303335049*48974153515499894645234965703965797304135752880534893557058100045319995640327414406010484894887092334290124777*12693500454889611206532256075665637940212171162010189388440729662267877275515941250815914080365061254677361638056776706610356922013384369276221
n+7 = 7*23*47*1823*2273*2447*2801*5517624133009231*298951889844775159*1583866351585911133838401*62186682346004723275580488729752311*1468062138781954877605806181091162927*17659000145579387114655129432669393871*88970230145880244982401312177602815659065896037765606484677123557913151049*6906100861906711042294651808315507604303950918775090357302270975396916428842528615127185377473*108812319169865611113499765242170244114701137148935942205033082816725856614419925622882473381377372631922944203113004898243458798930001960412143970055880623982022558497599
n+8 = 2^3*5*109*106877*271489*548347*1889029*690425747753501*7128686386107336359*61723043861290773050031488327281*327431229614131117772785108403020409*42690181228630617580877485609562883338265519449446054930706854035269082610743203*32324931823745846776555676474953076385481167075445075674853087885869075715823923787917*3364463967918569877412428970831620168291007106120572015124936370187375287372651445665862816100243713642209184736833502743289161172062635744244868288210173829656917423066932066331645933242315304125288886809659520629140738792514459
n+9 = 3^4*31^2*271^2*1213^2*290717660692056178627380230059^2*55183042568818223466137174736479621212999091619327075187^2*3286538828791738865769644759971695914014392529618376267704168559^2*50874028544198389096448371570878800888606150471855795807934093984638655897441174063482462519436790916281^2
n+10= 2*2099914577*969572514811180033060969*p488
n+11= 131*307147*5268867920473195353617881*p489
n-1 = 227*4373*1190082265745281*c500
n+12= 2^2*3*61*631*1783*88903*46906094056543*45467031493033267549*c474
n-1 and n+12 are unlikely to have other prime factors below 10^35.
There is a checked verification for k=12 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac521g.zip
k=13: 500 digits.
(n = (2*x^3-3*x^2-5*x+2)^2*(4*x^6-12*x^5-11*x^4+38*x^3+13*x^2-20*x-6)^2/16-12, where x = 2^92+227683166)
n = 2^2*3*1022689*11589211679*p482
n+1 = 5*5261*14307105795982965290513*p473
n+2 = 2*13*227*1889*p493
n+3 = 3^5*11*37*73*89*1697*22111*24379*50993*75403*294431*791899*7143163*37162403*40238771*318066373*774022187*1420502627*319986975727*360802459587989*4907888847800929*3948020931923766526433793457*1069158660766613132724682792996050143417*1026367444671582478750340776100892291660437*269849138487532374198877021349930016361247184220695675812643691485179934127024489200988726183*9828027147972763940840419206437324649030146749137361238797661494445753505040629992175660612970979552501716524157244914789537218745479849573839100489653483518323130703
n+4 = 2^3*7*103*167*1823*9436961405541853927*2442634599251757965491697450488141846145584325007794753532934341089*600199651586709792321245759194699649871225663594379892041934006939971309607*527494574388111009719149625221156839723560610238541931586701578366691138991354382424573311558118964044710317560296491439030482498012276207040809762647265709450370220697759850439666381328859614847517666066051906937892145477551456225050086697670164991720248902858139512339587289553072087611949432974971329045183361427759949669197151
n+5 = p500
n+6 = 2*3*5^2*29*2664932713*64053375527497*12950177265791031072639356885920439*34173198741742332396716976271692371701489770814403893768924387440612462806903842625552319985422420268803792535665370032811*38999042013591597497050209246905972783624407330496158969655866746662086214865227946049360131542356116951118562262174103446007490605710219381435049081471459942844240208575133436379007979279145337209263917682421878854593870018670730188222715372117686044826074644600648361727317428823657930132716062226843897099648501007
n+7 = 4001*33739*p491
n+8 = 2^2*829*26342725653094583*1470343579963822912245711071*187007870852441168222504281550299*99610574090928502236269502589968168206515652266513541*113895390854000729877269012409350207774928221694641996544761053*4913497600365917962155071410041034922653694031638944416398503899346474340572860479110111*9571423313604982533704828521048066928977532586908807859259654574404430464964151003988095630328625510537874340855221392615775029906318151824100974216194570356545874213060957042918269033449215300619413721816385325826807
n+9 = 3*61*1019*14243*52883*269749*73255439*1462827239*7462473053081018587763*1071679548735911780408902157698216943*36445304291066947549904502068646535721336497*484092028964579690545000872590192774860759892031294832503076409560989933338876408875527131*50995613413068475078730791162445890545117220641170507071916755522357759495828448552610940703376080579233279139258127608797223*438679328453087683131899883194844964328056296306402874587431956843033351755942263661835474320624975100141975738613328658670284212164843893047348661
n+10= 2*31*71*479*601*2521*10079*11351*5160785634557467417755921932864257*12956653104462928747086439183606463*2976475056377687756545777040724081181425791*1111195533664096264407102983373638795453449159*1411236536953344109648201030313903202292715855689198457423066135444942202534466870230539797279*1202453802380202612900570567249140337265977386730165272588662040901323849540240433058583909567403721087774087063*93428370649115500684161484486657066107657288857729057476468058018820851220113189554394219484855663733342770455665657679
n+11= 5*7^2*19*2039*2203*18917*20773*128663*206197*337283*4205437*4253763823*10361106142511*48219445919633*38944767299226517*823146395611037243*2567947475640367903*1980406199705343784496335146833*4068205927282173242680543824629898847404490879*170724646709285476974463174780574714342378046069994801064788497187739411448501668215598461179709025266247*172234777294748971503064594510341553906830528975589697516125816488729556569330134189448072867155899116079219277251511588963969376789767369662120332326617391898100471591557566930797914101561
n+12= 2^6*3^2*41^2*59^4*199^2*3907^2*1067282959^2*90695686907824373614544935809648303855581556170634564267689^2*1045590928167366905609128138910595915667531392289262793343319247317^2*13281367071696254893447093955890080700243546027305640807619578081114123230285603623403537002718792146591^2
n-1 = 7909460491591*854342030873171*c472
n+13= 17*c498
1700 ECM curves with B1=1000000 and ~500 curves with B1=3000000
There is a checked verification for k=13 in
http://immortaltheory.com/cnt/verify_k13.gp.
k=2: 23249425 digits.
n = 2^77232917-1 (Mersenne prime)
n+1 = 2^77232917
k=2: 22338618 digits.
n = 2^74207281-1 (Mersenne prime)
n+1 = 2^74207281
k=2: 17425170 digits.
n = 2^57885161-1 (Mersenne prime)
n+1 = 2^57885161
k=2: 12978189 digits.
n = 2^43112609-1 (Mersenne prime)
n+1 = 2^43112609
k=2: 9808358 digits.
n = 2^32582657-1 (Mersenne prime)
n+1 = 2^32582657
k=3: 200701 digits.
(n = 18543637900515*2^666668-2)
n = 2*(18543637900515*2^666667-1) (2*prime)
n+1 = 18543637900515*2^666668-1 (prime)
n+2 = 2^666668*3*5*43*347*16785299
n-1 = 3^3*89*c200697
n+3 = 17*111767*121224109*c200686
k=3: 200700 digits.
n = 3756801695685*2^666669-1 (twin prime)
n+1 = 2^666669*3*5*43*347*16785299
n+2 = 3756801695685*2^666669+1 (twin prime)
n-1 = 2*c200700
n+3 = 2*131*c200698
k=3: 100355 digits.
n = 65516468355*2^333333-1 (twin prime)
n+1 = 2^333333*3^3*5*13*37331321
n+2 = 65516468355*2^333333+1 (twin prime)
n-1 = 2*7*43*347*91579963*c100341
n+3 = 2*11*163*63530743*c100343
k=3 with a prp factor: 100355 digits.
(n = 37581121569*2^333334-2)
n = 2*(37581121569*2^333333-1) (2*prime)
n+1 = 5*13*73*5743*2342706941*prp100338
n+2 = 2^333334*3*36319*344917
n-1 = 3^3*7*13554581783*14805287479*67956715507*2168351203043*c100309
n+3 = 11*257*c100351
The k=3 record for proven factors at the time was a slightly smaller twin prime which also had 100355 digits.
The above prp record used a prime found during the twin prime search:
37581121569*2^333333-1, found by Mark Simpson, NewPGen, PrimeGrid, TPS, LLR.
k=3 with proven prime factors: 58711 digits.
n = 2003663613*2^195000-1 (twin prime)
n+1 = 2^195000*3*7*487*195919
n+2 = 2003663613*2^195000+1 (twin prime)
n-1 = 2*23*173*3863*1954173900202379*3612632846010637*c58672
n+3 = 2*5*35289796219*c58699
GMP-ECM with B1=3000.
k=3 with a prp factor: 74595 digits.
n = 777*2^247788-1 (Riesel prime)
n+1 = 2^247788*3*7*37
n+2 = 11*754121*prp74588
n-1 = 2*5*298897*17787571*c74581
n+3 = 2*19^2*c74592
k=3 with a prp factor: 74288 digits.
n = 1363*2^246767-1 (Riesel prime)
n+1 = 2^246767*29*47
n+2 = 3*5*7*13^2*743*prp74281
n-1 = 2*3^2*6930977*131315467*70439012053*c74261
n+3 = 2*834797*174218530427*c74270
k=3 with a prp factor: 67218 digits.
n = 2347*2^223281-1 (Riesel prime)
n+1 = 2^223281*2347
n+2 = 3*5*197*1213*1783*prp67208
n-1 = 2*3^3*7*17*431*2357*45611381507*c67198
n+3 = 2*139*4153*29183771*c67205
k=3 with a prp factor: 64868 digits.
(n = 3045*2^215472-2)
n = 2*17*1151*27067*173473*prp64853
n+1 = 3045*2^215472-1 (Riesel prime)
n+2 = 2^215472*3*5*7*29
n-1 = 3^2*11*13*22091*417293*c64854
n+3 = c64868
k=3 with a prp factor: 60222 digits.
(n = 38602791*2^200026-2)
n = 2*(38602791*2^200025-1)
n+1 = 577849*2645749*3427009*prp60203
n+2 = 2^200026*3^3*1429733
n-1 = 3*101*339541471*c60211
n+3 = 5^2*7*397*49523*c60212
k=4: 10673 digits.
(n = 245363571*2^35426-3)
n = 3*349*111602773267*953666301013*43440278284896679*p10630
n+1 = 2*(245363571*2^35425-1) (2*prime)
n+2 = 245363571*2^35426-1 (prime)
n+3 = 2^35426*3^2*1117*24407
n-1 = 2^2*5*720418081*209358848887*c10652
n+4 = 5*7*17*67*137*c10666
74 ECM curves at B1=11000
Primo certificate for p10630 is in http://www.ellipsa.eu/public/primo/files/ecpp10630.zip.
(This was originally a record for k=4 with a prp factor allowed)
k=4: 8193 digits.
(n = 378149751*2^27186-2)
n = 2*(378149751*2^27185-1) (2*prime)
n+1 = 378149751*2^27186-1 (prime)
n+2 = 2^27186*3^2*7*17*353081
n+3 = 5*1019*24923*43651*989239*p8174
n-1 = 3*53*127*7043*626921*24117041920337*c8166
n+4 = 2*11*561139037952529*c8177
74 ECM curves at B1=11000.
Primo certificate for p8174 is in http://www.ellipsa.eu/public/primo/files/ecpp8174.zip.
k=4 with a prp factor: 10043 digits.
(n = 21996007*2^33337)
n = 2^33337*11*29*53*1301
n+1 = 3*5*1129*prp10039
n+2 = 2*(21996007*2^33336+1)
n+3 = 21996007*2^33337+3 (prime)
(n+2)/2 = 21996007*2^33336+1 is one of 28000+ primes found by Ken Davis
during an AP4 search (4 primes in arithmetic progression). It is a
Sophie Germain prime since n+3 was later discovered to be prime.
n-1 = 13*191*1879*13153687757011*c10023
n+4 = 2^2*3^4*7*258653726077*c10029
74 ECM curves with B1=11000.
k=4 with proven prime factors: 5638 digits.
(n = 25390425*2^18703-1)
n = 7*15569*150611*1338793*p5622
n+1 = 2^18703*3*5^2*43*7873
n+2 = 25390425*2^18703+1 (prime)
n+3 = 2*(25390425*2^18702+1) (2*prime)
n-1 = 2*4657349734574297*c5622
n+4 = 3^2*13*47*10528525069*3236067162807210923*c5606
221 ECM curves at B1=50000
Primo certificate for p5622 is in http://xenon.stanford.edu/~tjw/pp/p5622.zip.
k=4 with proven prime factors: 5257 digits.
(n = 297079965*2^17434-1)
n = 4363*22567639*p5246
n+1 = 2^17434*3^2*5*7*13*72547
n+2 = 297079965*2^17434+1 (prime)
n+3 = 2*(297079965*2^17433+1)
n-1 = 2*4846109*10719641497868929976071*c5228
n+4 = 3*11*c5256
221 ECM curves with B1=50000.
Primo certificate for p5246 is in http://xenon.stanford.edu/~tjw/pp/p5246.zip
k=4 with proven prime factors: 3500 digits.
n = 136857*2^11608-1 (twin prime)
n+1 = 2^11608*3*7^4*19
n+2 = 136857*2^11608+1 (twin prime)
n+3 = 2*31*24337*674501*23226188779*3840896415363899*p3462
n-1 = 2*5*67*199*14683*171937*541573967479*c3474
n+4 = 3*5*338817126533946144966094117*c3472
220 ECM curves with B1=50000 and 140 curves with B1=250000.
Primo certificate for p3462 is in http://donovanjohnson.com/primo_cert/lcf_jka/p3462.zip
k=4 with proven prime factors: 4187 digits.
(n = 240819405*2^13879)
n = 2^13879*3*5*3209*5003
n+1 = 240819405*2^13879+1 (prime)
n+2 = 2*(240819405*2^13878+1)
n+3 = 3*13*43*358877*p4178
((n+2)/2, n+1) is a CC2 (2nd kind); a length 2 Cunningham chain of the 2nd kind.
n-1 = 11*19*11405214139639*c4171
n+4 = 2^2*1123*9461*133900079*4881536249857*37975791341*3460283980865333080373*c4126
221 ECM curves with B1=50000.
Primo certificate for p4178 is in http://www.geocities.com/scooters_primes/4178.zip
k=4 with a prp factor: 6223 digits.
n = 2989530439*14489#/5-1 (twin prime)
n+1 = 7*1123*380299*14489#/5, where 14489# = 2*3*5*7*...*14489 (a primorial)
n+2 = 2989530439*14489#/5+1 (twin prime)
n+3 = 2^3*943127*5020192965913*prp6203
n-1 = 2^2*5^2*47303*76425929*5735860488526391*c6193
n+4 = 3^3*5*c6221
c6221: 20 ECM curves with B1=11000 and 40 with B1=50000.
c6193: 19 ECM curves with B1=11000 and 42 with B1=50000.
k=4 with a prp factor: 4187 digits.
(n = 240819405*2^13879)
n = 2^13879*3*5*3209*5003
n+1 = 240819405*2^13879+1 (prime)
n+2 = 2*(240819405*2^13878+1)
n+3 = 3*13*43*358877*prp4178
((n+2)/2, n+1) is a CC2 (2nd kind); a length 2 Cunningham chain of the 2nd kind.
n-1 = 11*19*11405214139639*c4171
n+4 = 2^2*1123*9461*133900079*4881536249857*37975791341*3460283980865333080373*c4126
221 ECM curves with B1=50000.
prp4178 was later proved prime by Matthew Peets with Primo at a time where it gave
the record for k=4 with proven prime factors but no longer for prp factors allowed.
k=4: 2135 digits. Same as n+1 to n+4 in former record for k=5.
k=5: 2158 digits.
(n = 7096755082*5021#-2)
n = 2*(3548377541*5021#-1) (2*twin prime)
n+1 = 7096755082*5021#-1 (twin prime)
n+2 = 2*5059*701399*5021#
n+3 = 7096755082*5021#+1 (twin prime)
n+4 = 2*(3548377541*5021#+1) (2*twin prime)
k=5: 2158 digits.
(n = 4790484140*5021#-2)
n = 2*(2395242070*5021#-1) (2*twin prime)
n+1 = 4790484140*5021#-1 (twin prime)
n+2 = 2^2*5*13^2*1417303*5021#
n+3 = 4790484140*5021#+1 (twin prime)
n+4 = 2*(2395242070*5021#+1) (2*twin prime)
k=5: 2155 digits.
(n = 14635080068*5011#-2)
n = 2*(7317540034*5011#-1) (2*twin prime)
n+1 = 14635080068*5011#-1 (twin prime)
n+2 = 2^2*7*19^2*103*14057*5011#
n+3 = 14635080068*5011#+1 (twin prime)
n+4 = 2*(7317540034*5011#+1) (2*twin prime)
k=5: 2139 digits. Same as n+1 to n+5 in record for k=6.
k=5: 2135 digits.
(n = 447295839*2^7061-3)
n = 3^2*5^2*90439267309*152397278879*p2110
n+1 = 2*7*11*17*23*31*43*24633501044437*5238873869796767202503*p2092
n+2 = 447295839*2^7061-1 (twin prime)
n+3 = 2^7061*3*347*429679
n+4 = 447295839*2^7061+1 (twin prime)
n-1 = 2^2*271*4273*208933*567367*25463453019262592726380909*c2092
n+5 = 2*5*103*15241*c2128
226 ECM curves with B1=50000 and 120 curves with B1=250000.
Primo certificates for p2092 and p2110 are in http://donovanjohnson.com/primo_cert/lcf_jka/p2092_p2110.zip.
k=5: 2063 digits.
(n = 1749900015*2^6820-4)
n = 2^2*(1749900015*2^6818-1)
n+1 = 3*173*2707*7207*p2053
n+2 = 2*(1749900015*2^6819-1)
n+3 = 1749900015*2^6820-1 (prime)
n+4 = 2^6820*3^2*5*17*37*211*293
(n/4, (n+2)/2, n+3) is a CC3 (1st kind); a length 3 Cunningham chain of the 1st kind.
n-1 = 5*13*759476989741*2089473326209*5861811536579*c2024
n+5 = 7*19*51610785291067*c2047
253 ECM curves with B1=250000.
k=6: 2139 digits.
(n = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-3,
where b=5026578700, m=13416739015680*b, x=(2^67+28683395)*b, y=(8*x^6+24*x^5+50*x^4+54*x^3+41*x^2+12*x-148)^2)
n = 3*5*115523*2364592753*16987001959*6805713408228112745527499*p2088
n+1 = 2^2*7^2*462899*p2131
n+2 = 191*229*277*317*353*421*509*919*2243*5879*12377*36901*76607*80471*224629*266059*329941*380951*904919*2136311*3056057*5914483*8096183*9465601*21550513*94799377*130860217*155882081*45822412859*8146655372227*1474846703267549*14278046242057967*30669562031647339*50840461718040961*92756515760205629*52290238661226907927*96677437656827057417*1024366806097559014891*11420007000929879319421*56573337948259077277921*1112138785081162287658151*8528010785159544755667973*36811892890610855550478433*1097776113917720517099045197*6266573674771665058327422803*30451289066171237974920654932085755371679*1316506222346661036532477000837519412493197*111403796509772263652769724399362571727608328017*3624284692109954237076203309579972669010531411749*231489152760278164551457534708931642648344475815232745703*147116374129195711393035548788265178287722577822324906900753*20859830985834684423322955342287747410510371603352932446294829*7904406721787550323058091000193526797479128058982617278101662873*46540398168652715463731610192037761520603950804914554357304827651*257704475111226397986479160737832082626014496927642276922664061017*1633827832645400817312925190638479419466129452872018361627854113293*382662618255823283472179277788367200218513195283130424126700215295748735665441*1391507993148700240606140015038276134257033549860873095114389589694682942562789*7499469611937904980182356228537145717109317190667055527703147049893120442453955199*356484869506390402253567489057105023111650744085980620919627955814007320376395156783629*4914377652267665655490741867052323842264433563348177699667952493668432484240598433717557*378510207963271339787785564166177590272761189694519309918631082645427918103057025277821365211654046232903*10882730321922719433671807500165889524951973748667067180984450031864886847971033873122407056582166336612089*42492274401367782319359911698875554225735898020880648763949946800237401567377284028369105348871780497889130827*550436717882902423471815618746320309019776404049358761651392910282277600444836241713723132473766354880050594580270621227616367362518126662367*5114238400333688448457232468331562020643807865101179717783965219897043019763899891052712357151409461393105479520684052940760427498538444060680207904388052821822493913325171
n+3 = 2*3^4*11^3*17^2*41*53*71*89^2*131*139*211*251*409*593*1277*4583*9769*18289*49019*54371*67079*167381*286009*463717*10388629*17815571*66515707*698686561*818125109*831161249*846265213*1212778709*2355934153*3058330189*4150299187*25306735499*3604486352897*14948583710353*23141076969449*153460724640233*334741271181311*512852123180983*1055641941839291*1392151174305467*8867626867246039*9667517060163091*182124553316524501*202781152357277099*406336869368999861*24740829299338719433*29322679844588024497*24294379584087164777131*9581091219467836447210057*11623108404437621322717169*680338395030015030231632893*20518929682734143207132735353973*66563637308857605625389901982459*232385937495200968879386754974313*535735630734461728406216081733553247*508439874253891778606279503653232082681*568792881367546646342582771622531398201*8858542841440374529189506450521929269739*10843913415694740420736960889105018295370049040103*256316245276700640596341494983955113716708347284713*269332371965553888044556108484090910220885368743111*33712744122427700771320939883696582781318705265620304873*485185740258747937662437489843040968838887833679684367157*788833359586429565859990494779597014569532914058188795561821*7291571476603897113533712637384336734903709885014047061381079*19669397718658187975325767914206200463853657104127548035301433*3125382489664258590391298530427593160788815853502138689793042677*4328160415203489875259678235637742289270192182210119710370643850927*309822335403073024796762355934933497647408610442517613113885861757379*1103898050573311594319428118682109119879937737776005655410652663836349203752737*92486996234402935152972092853138208762410572617958701845300321611350478666752372301*500009341040688147102544811636097201842624578131324143223416593271893196262498696726469*63547811132003317055239013988352688043328652147686430763336739113148789999768204463467171793630031029*64319007646797325362848402230487990288109209313701267028024171110636741054514980740206560574125978219756541130779571*842856784319760594663011094715040960562959489469298103728654570968227035118953458504428238545469047549975587147482030615502207*37629800802773044149485257881001665650425553616023615308422357601471426709159760613987492291949078144731754876614032040655663376557577609451
n+4 = 13*652739*p2132
n+5 = 2^5*5^2*37*241*28843*794200385809*p2116
n-1 = 2*6911*18959*675413*81134281*c2117
n+6 = 3*167858252466989944843*c2118
n-1 and n+6 are unlikely to have other prime factors below 10^30.
There is a checked verification for k=6 in
http://physics.open.ac.uk/~dbroadhu/cert/ifacpte6.zip.
k=6: 1803 digits.
(n = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-4,
where b=4011209802600, m=16812956160*b, x=5000001251617*b and y=(2*x^6+3*x^3-148)^2)
n = 2^2*71*173*1567*424064851577*4764539601163723243*p1765
n+1 = 3*p1803
n+2 = 2*701*19079*282279559*p1787
n+3 = 37*43^2*59*61*83*89^2*193*227*353*563*673*773*1481*1553*27617*37633*58147*64951*205201*402769*435569*6568097*8204233*62792377*257777969*775643203*1687171501*7901714521*8611020113*44900871599*134596618103*194727326069*4054656560329*35310788241601*1144536016916393*547418495121864799*960543818610206749*7512682467497170148153*77122988632197567410243*1041998597004029214079279*3934208076568895847863441*76355031909623483424534049809013*9541464799334181477034188730576609*40869923748552947588194499753142151*173375138423053317576233169105066161*24960491263045151537938246748797732033*4046241609244762139507273437307875056559*934101794602481559648481462842733450340177*100820663465058033355399489727414672691672458489*10355778250337305458080594000027129114245922478341*157937875196459568563896915849429921595309413943803*35332528502521473760795720289180986110801399084551341*732696498696012722939913820432299305647881186929419241*208640843623123680312945317256315243660831682858757623357*76201886634791577171993998077475484980646225736074267481987*6438866769945748412687984544492447585022109014683560792252689*34210709690822198267146724151373988688494165650865775297103199*2132791641418224620482034613867224765473751663257207506429723889*4874057484841525794516697638376051106947984422695239381249223971*13560606627692164809189162883273527153385704647924894629000192913*3352506997242646456006759979730890421610858650126457579100616082691595385831349*102692044298884915205557323891340757406411668602429945398737447511025039561640393216003519727251263307*189922589414865226468622828424540278517145146779251204153423511423539426305694130190815480795318828893530509*21266168480080580156785583576426010575615342816000905263892120638719521665707623169105733601117777886837039611*55149006508549405130037208376035974596763660808047405369568008279354529871030586217152406508249456842538207077307963056464229847
n+4 = 2^4*3^4*5^4*7^3*11^3*13^2*17^2*19^2*23^2*29^2*31^2*47*97*349*677^3*1601*1741*2917*3331*4229*9661*10193*13789*30853*38713*62473*150287*186239*646981*866293*917101*1698043*3746179*11497231*24666283*36191671*49429327*50339537*85334927*124264571*860249329*1288198913*7385526221*7385526221*7385526221*8477874479*1141359939089*31077751869103*326416583825723*523883179531643*3571319405698511*24185665817420131*33548677916339821*225006287244512797*395997081191384419*887523182980651073*1139672144933035073*7667770680246290221*23897473911488134597*1100989909387778738903*2625800907208358841833*131639598063226658510639*715781480481372739220274931*1180722659058952538240342286749*10685698868228397151327332353471*14460423523793974387623255510263*411857497472623648085321963833273*10876101946069142809741344894091519*16981907479154384669308748833397867*22203951185157442583966047769588057381*5402898330595406519134147306421071728281*231491914441581992784465543486871431668863*1025339487917652994806865866045830130341031299*31115421453323513037476889813038334113201214677*67804647002031445271110159102947426615620185783*12603537558996293938258626344198230911354106273722673*553954117005392087403276328493242186322972372392295239*4884969711235714491445156179048143737630467138460814567*2359583141365393112820375931702887501111092275711062284863*21683720997503404786034957307842654904006922132535428397388587*1194426526170945773717097618626048988920226654130164934661119087*3179321305537807099689388278906670933944837646065351852662778017*2970796650255750198546518165137105392831427224063820842847779446318121509591*40550699121998067419991842732804823585659593996219670981404869979254505478981432765437046291*68698278844643201913685198175034756270603298277530436699827937740857333506176487316952783366947983291613913*64775450186718835923061966374498704041978953377071501172637808318170164907715379477552698465322868941229874726228423297084374755363
n+5 = 419*3407*p1797
n was constructed such that n+3 and n+4 are each the product of known divisors with
less than 154 digits, where the large divisors are suited for SNFS (special number
field sieve). Such divisors of that size are possible to factor one at a time.
n-1 = 5*239283371994839*543315317127619*4154925272246149*c1758
n+6 = 2*8081*33697745047039*268690037677741*c1771
400 ECM curves at B1=250000.
There is a checked verification for k=6 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac6nfs.zip.
k=6: Second record with 1404 digits. Same as n+1 to n+6 in current record for k=7.
k=6: First record with 1404 digits.
(n = (y-22^2)*(y-61^2)*(y-86^2)*(y-127^2)*(y-140^2)*(y-151^2)/m-2,
where m=67440294559676054016000 and y=(m*(10^96+9581328)+22)^2)
n = 2*p1403a
n+1 = p1404
n+2 = 2^16*3^10*5*7*11*13*19*37*61*191*227*373*631*1009*1289*1811*12251*18223*19949*39761*40099*124139*156967*452077*2130173*2705537*3462937*7633753*8515597*33173939*52641283*56707439*2678042047*6028890499*235057646869*2293399729639*4716162386153*20290714976827*76966210128923*82930514647957*3414960948392327*4798235010101597*492989197272845783*1021070851282029949*346642803549673938493*10381965549080973912121*10705853500705134572111*3402602170075407160915063*3896113165479453193941681514329473*572982916505698266772046961556571719*7779627119389705894268583790178040713*2679272214955462246938889432925080340640361*49177705726337585346837755875445232529608396349*241826146812234755480438342468089149152034499363714121208377*243192829165886509367881231894318437733436233104128418423890411*28990423033538119642473011651390774174058997738460601175798278583*437141645229051532697099686819531598452795793496929224693672331927*1418753969662679659418666553265322031463146093384848016223510836379*645413449365286314824919986919822970362519337008974665550190920457255518759*20965538503194242905032903697401432404999762007708204375118749419811954970647961*3294297702148718077561555791777335845260219149060777243934402094257398262227750093*2406629749273883289769692163217837087029221186833486436482263322161068612926191024481*234004321348347878243450131076080120876538597870136557022678611648262729687220946497966699*2342885206975304600030966940136109235370201749576148604770764630664574779225197114034373688176651
n+3 = 89*137*149*257*389*541*853*2591*2851*3331*7559*75329*93229*102667*225829*373631*1307347*7780769*13541653*99379921*188069213*226248731*1457666107*7019523803*69270603191*130601553923*471637135901*499457115919*6996512652487*90468395140231*126512656915647001*561436143437641939*39622341152320502977*5060294951719365191399*15507396773627356960691941*34802122844262613744836487*73113052821679897631882999329379*13881170525247452115444724778498657*11236972040374240265121033071750573780554937177928509*26291144543042185831481954556908348064015228448587607*5441441819431723433035024498287323261369539563023632649*3323253180187898782557645135868186230115978698950734843531*670187939956550896144016063406622725370069384221626353600796980706687*388508861296593678793647742126412846197330928103032998798526196672169751*46526361176976331131692448831886295654180937200509756317956476079377060276107798251*123723846994132061509262488616575553842266557999262274735157745365879370310957511849587*52777309311666259876826027372783385850666230122869905057157388781104477582284263792304493*214203286249223750965285191327202298873780724143364323737823298751073982098758638527694627227*30091809213543725941251060360120799012501436121378265374301342522682366334233176762803248560731781*1444362720210768757829252247484379543219331174080222949691438360399704398339766729979542816990814772477899*560896070384991947831892486100383451139489111138563275709977404579309957444584330524226413933472530198395863
n+4 = 2*23*24943*13071241*p1390
n+5 = 3*p1403b
p1403a = n/2 and p1403b = (n+5)/3 are different 1403-digit primes.
n was constructed such that n+2 and n+3 are each the product of known divisors with
less than 120 digits. Divisors of that size are possible to factor one at a time.
n-1 = 3*59*104882036089*123707274359671*c1376
n+6 = 2^2*c1403
320 ECM curves with B1=250000.
There is a checked verification for k=6 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac6a.zip.
k=6: 1104 digits.
(n = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-5,
where m=67440294559676054016000 and y=(m*(10^71+145589)+22)^2)
n = 2^2*11903*8908713059*20605508509*171399027325457*p1064
n+1 = 3*1867*273269864239*3658698250513*p1076
n+2 = 2*p1103
n+3 = 181*p1101
n+4 = 2^12*3^11*5*7*11*13*41*43*47*61*73*101*109*167*179*563*769*947*1669*4483*8423*8689*86857*249421*352441*1452419*4213189*5337557*38376203*75807749*1342644727*1592856247*79036647043*2066440556016253*153467969439714187*1015711703948543083*351255950120842197943*1181227353539948611763*1748360760308591989432139537*24015455632265221840820445967121*80309621927907102740140515572868133*109269974651161535058599454239640944990321*9517480444270086129756297023952251070704479*11624759710856712926151094525593318553483268019727*1624383839750483864275832116445201199733452749965743041098763*3321388188935613335625081797063937564073265903733973310154511*3930918517802853429791414852699425685138947223854804684150991661917*15700457668245089126937307919442053598829465908914062668377242055044331*613509075233122056292153808622241583039288732180711585974780991303046186587*2259724066520182819518427246354597484635432626048626394948984886210453682509*20145661260561675391127046332451100091532791360010608280969949945015502216917*284317035570965203892357347738074193719696256560671468387518710886460130970549678393*750409068179835827663191219069403815613515074981151907320017458837177845836411021913
n+5 = 23*31*79*97*113*641*1811*2357*2539*3253*11383*11617*12413*460111*545641*22231589*30740459*158884357*346304197*9596667107*11982265853*172611395597*42341290844569*128209993016077111*233324021865808229*1067315559086224029308057*7756943116200250207544789*324725593355593676830982063*1518260580302799430092803189*14437709862395025185324753357*741946562546897516077231120190975401*1215947305584459125543131239649844633*2459003388630363360188816574544627010587*526712065096184394205028755707683695993381*2482021200509575214080292599399563875733646553*176252349918469581244355387429024446510384958783767189*829503790824119658573264002082945813894732504167507643704839870307*5114226172787067738433625217783426484550351898768325916811724886909*148389983113982515325263241221475679888839593884951677946444415461929182414531*55666846987356792972772284987772094350949848491397034769842185515267556045842978847*758457842480388689538457718990000165699372176981969017542726678962273839761971061113*58138184965237977600000000000000000000000000000000000000000000000084642802109040319208064001*518771496612892723200000000000000000000000000000000000000000000000755274234203744386779647999
n was constructed such that n+4 and n+5 are each the product of known divisors with
at most 94 digits. Divisors of that size are relatively easy to factor one at a time.
n-1 = 5*1948359547*c1094
n+6 = 2*11969*4471247659*460041067299095322777708682147*c1060
700 ECM curves with B1=1000000.
There is a checked verification for k=6 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac6.zip.
k=6: 1037 digits. Same as n+1 to n+6 for former record for k=7.
k=7: 1404 digits.
(n = (y-11^4)*(y-35^2)*(y-47^2)*(y-94^2)*(y-146^2)*(y-148^2)/m-4,
where m=67440294559676054016000 and y=(m*(10^96+10624986)+22)^2)
n = 3*820681*58279268867*8089381088597*p1374
n+1 = 2*29*3317521*15351437*13973807284874590379*p1369
n+2 = 1427*p1400
n+3 = 2^13*3^10*5*7*11*13*23*41*43*53*61*151*191*197*277*617*761*1063*2251*5419*7417*34649*129887*148691*215297*249377*292601*310117*333539*392647*2602291*5158817*9098161*9572413*46347793*47181619*49687879*91731397*116976971*444476651*16651689479*141790225477*11862804584870311*475827842069624329*1952556888901014284396298871*1656027221545269854046521732323*49640146225618621543893578601492354823739*8682665993637814126334655623002945126132381*788345137743082394797513912882652922339198530323534223*46431605934975407105850843188010367507618025283882564529*3366617260222806408173455918182918611428964447899257888247375663*2719605820033375780663668494243928674314151927435135054868093296655402493*69099219944925942336457370755387365575325867696295211021371495432795990129*7423954758405991178947245926584616929504622702291806044646152378579768926683*507124864910740617317834708384624322829685165741783347184747283863236535136436949525978801651*363525565401269080033916712603583884338729712584004041380896373487530555465536191928885837667299146959*852341347139314030568300485338245515621488696837146351125562512551910877500537794424305520419431727471996047*3558804019347548057129979882963459534270211185413580250765768246811588239876718982576493662356513224206328931*230485523151582031558333703575859275942324188912546436956811494150737693992843496775472403431813922482952001036373*54651778411406850904376012965964343598055105348460291734197730956239870340356564019448947096071463154322721600683773
n+4 = 17*67*101*103*107*109*263*373*463*751*6899*14629*22129*44491*234917*1146833*1859983*12292663*3489932473*4653304463*13228645723*56901587167*443097598171*2569113463417*6080736107339*205932998345467*240850667750249*509184985561331*6453823454403719*1545871005161244499*1775442683235787970389*2451694912860432213727*89792269186185778201159*1433951135487402561509092703*1476894477771977030320654201*7585718198499770808806342078801*281519710728123512693670143339827*2356521307346163098298477871916817733*52779008407884145440776317064241963137*6848307225277797080796031980759801261721*36978195796816375632920076819259670928713*5989972262101629030141145495263426612254919601*32113835899565282657830284793564132341737179238863*91609376828281820114635949438428467784649774584286839*36786583168467743260795894602757035563550833953360713366253*2075649148044335815479282359000619116345060219424797088278928237306672365791579*14817649457223344618882571026934739805193947170804011001801725368635753671216882070398683*661791067702497042286854813438731762743946616995889032833714288555760467878168944873689801*12113030945818324826368053939966187535827171168662412479040300135359529000321647451437672811*174608865305212009166718383113652644062482549528644096958004773623125241833539831576705915580917*486425920865949537489015750397803498829412176281597674228671188335610801241814848467600911390599*2823160223584328544758003137642306989810953614879045695865510201460912967585227792523246402587435978803
n+5 = 2*p1403
n+6 = 3*26535329*37907509*1185659957*45446838221*334012213438151*21633422296379581*272780466084810349*p1320
n was constructed such that n+3 and n+4 are each the product of known divisors with
less than 120 digits. Divisors of that size are possible to factor one at a time.
n-1 = 2^2*37*113*988979*2759209353591763*2338830543576644077*c1360
n+7 = 2^2*480023*1408111*32195489*5342864083947636432359*c1362
400 ECM curves with B1=250000.
There is a checked verification for k=7 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac7.zip.
k=7: 1037 digits.
(n = 21247003564*2411#-1)
n = 12906420959*p1027
n+1 = 2^2*103*51570397*2411#, where 2411# = 2*3*5*7*...*2411 (a primorial)
n+2 = 2524541*p1031
n+3 = 2*5002841*6245491249*p1020
n+4 = 3*485475518243*p1025
n+5 = 2^2*(5311750891*2411#+1)
n+6 = 5*3691*23063^2*2961991*19076087*3778442561*p1001
(n+5)/4 = 5311750891*2411#+1 is one of millions of probable primes found by Markus
Frind and Paul Underwood during an AP8 search (8 primes in arithmetic progression).
n-1 = 2*63501029*1676569836074094735760183*c1005
n+7 = 2*3^3*44041*50036011*c1023
453 ECM curves with B1=250000.
k=8: 804 digits. Same as n+1 to n+8 in record for k=9.
k=8: 703 digits.
(n = (y-23^2)*(y-24^2)/55440-6, where x=(887040*(90^12+111833012))^3 and y=(x*(5*x+9)/2-31)^2)
n = 3*p702
n+1 = 2*11839*46499*87049*7446793*12390250603*p672
n+2 = 17*19*41^2*103*137*157*163*241*269*449*4079*4133*4721*4861*10753*2351263*2485073*3057007*57482213*68219878477*381488038681*462620692403*12683050903123*411332105162957519*3276055668231296267*6870702102249368659*9669550233544435375557769567153*2360578532615379194929178201025349*3892682480172514970480939619480019223*10494347741384920661144877561340083568978429272003460742633*1829055837237982915251619336597180975905571357734604112099369675323879*9867819978384278619956294537368978925192159544086646398961374957292063727*3935161077451634316971150930230328910277288574512329708155729392891380404108714093*1153403655532048033020388150472523156118899935315742577335109467337017712352307759295113803534362481448755802478888663421616880604044232388184863656791
n+3 = 2^26*3^6*5^2*7^2*11^2*31*59*83*89*239*293*463*1237*7573^3*392443*906823*981742453*140477207564839*9323568482800743161^3*7795461012843187990059199*367976472250843280791966711358177953*2369898157201664369071987219354981398749094340205776257348069340155257087*8586003417616576718090324916324707595976594797359752098641043332748650167*8735493552188193449423855866811330500338440890721691851514216931005883399263423037440001*3170970794225727329321200288517180338107872852190460711796341512855639339227445852183298386837309339215974029658316649854721145406061361*23986911952889385545207942919750495805093443987293237095248265406287411769986943478571076469349420621340301274180158830039251291653843647
n+4 = 53*173*p699
n+5 = 2*257*16001*p696
n+6 = 3*13*23*487*641*677*1319*6961*55609*70507*116293*274973*392159*14237603*68584667*303775457*425585317*13728113437*131679928339*9654557976929*12435367768589*630259423347317*901218628584317*164149758593811967*1988600182502075783*33149325702335458851579341*58130123165064994906452158476554824029*15223909874348358974851188325731198722002409*15108855722066292086389891853338191325033709928096121*499524512241167061789413229003729509184108096802211073409*42823638155592295501589956561528346972325869068174232243623637347*16951154355788530178104217461024513706300145304574366796538557804121*37680223587148534204317012443876372557929839640420906180155846961328886566664252556913281856156597565057740680394042206583621688028269799636710358518080817004999965407
n+7 = 2^2*197*8429*313933*260172449926811929035350826091*1269872752096078507979335383599*p631
n-1 = 2^2*199*66067*5519123*24972481*c681
n+8 = 5*2993423*124255771489*142190675346823*c670
700 ECM curves with B1=250000
There is a checked verification for k=8 in
http://immortaltheory.com/cnt/VERIFY_703.gp.
k=8: 600 digits.
(n = (y-23^2)*(y-24^2)/55440-6, where x=(1320*(10^22+1932187))^3 and y=(x*(5*x+9)/2-31)^2)
n = 3*7*12203*17669*1099168540978153*p576
n+1 = 2*778763*p594
n+2 = 19*47*73*151*163*1039*15619*92233*4891127287*22577760473*47111633003*12070429199483*199006571381687*505906662595487*1268153950947427*3112861189877545933*5854318051903813309*8967446026650440921*412153717834508405396923*2152209086997284753405165939621*37831708440169494583827256188208582483*8249904673643870473934306022816076007428663*9739353386493274920005734938730781351211448586959781*2446244936377136793664896517135503616925585329882636307*275172535645132588474698760948412898025635380628964475577089*57790582804607151344983828096115383604868580390780704722813738821*1671556500708834606767822131230920740407412585120283129906232745727191
n+3 = 2^5*3^3*5^2*11^2*13^3*17*23*31*67*1063^3*1055933*2582369*112864327*120841073*943018301*4214057027*26688059579*159090509027*267473033453*41857703804036059*723641363340328673*723641363340328673*723641363340328673*201583177283742768350363*6870502442454265018806723641*27596850040722511032814742539*355792854625524465962588947535325171255860024510489693551*896071701079444588332473182556959050185892140360842237573*484882078319894128785085629548448989511694520807645981942364330906558109195524376307681*6262060188743043039415010743222646616899150266572151851834807968161857158595536784536792596382842491016647411358484563906157076210321227747973493
n+4 = 461*613*p595
n+5 = 2*2621*p597
n+6 = 3*107*757*4909*6229*16447*65381*808187*177513353*296694881438759*10622943310153609*144830170731359871913005718268119*13994754697828882745886899442981334542757579*261939634522000611741702503697485328141259777378173867459727*5749920000000003332976202512000643993328980305415877184611415643348193760007*30575759654977077510752353461475337956997642899877800633700100675711794525297187617*11413913219685676030478828514755884493375522838318406507453170587599985178733468847520565119741918049927971765982974521*143338764107655986839549495454178765826740836571371361424355522329730096011609356106622869993752761596283471508374706064109
n+7 = 2^2*7*643*59407*23355487*p584
n was constructed such that n+2, n+3 and n+6 are each the product of known divisors with
at most 152 digits, where the large divisors are suited for SNFS (special number
field sieve). Such divisors of that size are possible to factor one at a time.
n-1 = 2^2*25711801*125018686680668134578227*c569
n+8 = 5*281*c597
1040 ECM curves at B1=1000000.
There is a checked verification for k=8 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac8nfs.zip.
k=8: Second 509-digit record. Same as n+1 to n+9 in former record for k=10.
k=8: First 509-digit record.
(n = y*(y-23)*(y-41)*(y-64)/55440, where y=(13860*(10^60+720251))^2)
n = 2^4*3^8*5*7*11*13^2*37*43*103*817793*173899823*22221761707*37836465211*1128969422473*7226426977433268109*666555585664696017970179047*666555585664696017970179047*4274216449873110505368747152083*4274216449873110505368747152083*924544586620285060733647007145063151*479490073146871050534575062193704807941737557*3752992276532013774050014526575133031229811053*7291083234967253095183997555458638002935444601743274397222687965174503227040871*228474332808418653912215340656711947252683332494996254331989692206925564018759152735292109921745527382957201
n+1 = 19^2*79*223*17021*129341*187387*470077*1134503*27596381*591071681*219722506332743*823024421902436629*7217017062935103051815890914354049*61551757785499981066434091025664731*62152466367713004484304932735426008968609865470852017981985107*149767418497179338435882188243580843563124341953736118862837592294108775607496189*11456039860137004758584981278594652001673176571216011518385988939472768725293749566253855701651241969269*125637734984402123698410978366257187832009797939597866455282183288466661002745093507460307485903514650852713133199
n+2 = 2*269*2647*5839*p499
n+3 = 3*77863*p504
n+4 = 2^2*349*727*1579*14407*140123*144569*419527*1125823*280638781*12482765623297556507*293244109006399769051*3924302134130214471389*37094900399021319425878133*8628414443325319794582933283*22394618831576964656662185799*85606406273330155008249155057*373302185646651554489812114237*1523737310190931395809246034113569*81026239243914061907529877840939545661*7541626939748321863588285645973660161581613688604461*43384942263778101127544397295737593104298414085317202239*2980895151661693476335327178562617958496303370683854299520305985770282863620140547091
n+5 = 5^2*17*2895637453*21087042971*35055489406115209*p470
n+6 = 2*3*2879*36931*p500
n+7 = 7*1823*38167*331906427*168969463248297611*p474
n was constructed such that n, n+1 and n+4 are each the product of known divisors with
at most 128 digits. Divisors of that size are possible to factor one at a time.
n-1 = 193*34849*44089*1520083*2478731*2289665947*c475
n+8 = 2^3*902141*1832165329*21321528412173053*c476
1750 ECM curves with B1=3000000, B2=4*10^9.
k=9: 641 digits.
(n = (y-3^4)*(y-2^10)/55440, where y=(x*(5*x+9)/2-31)^2) and x=(1320*(5*10^23+7574922))^3
n = 13*31*73*593*757*11471*12487*17041*54949*278029*901249*4790389*29063737*216219313*480617453051*730400840441*54101370630581*33174660281920153*1200313583779857538637*2021944779729779239081*2837368867353968687549*1061625261781019653637690586250321*19340092075750156710130380623411729*2061984639973675784172435043748808101*156392771603960271036296396091376763615503*2526909010888691858842287468626436550362106081567*3724660798182483488936637129340557412622657092030884777136438810213*71874000000000003266639662968000049489081298177777241917287123570268565196416001*126651843875397099513662125701570217778163140409122891242823544251509932501420396712305178111145609544574982476513345693771
n+1 = 2^8*3^3*5^2*11^2*37*89*977*2393*132647*41511931*99852207041*587299388795151066599*250000000000000003787461^3*1778007979566865640487680539*90713012204379285603476979477401*590488790619001905875488546671218429197711*17805009757329579532871514385178076366829311791*98905616340494743760280350646391078198892638531*71882161957120343138113743329439944242294214201425931996660692604102233*31757568132349259959665435705861571851253708159452840258558330447880606873458258600929*105749680163766642161165164223625010949665222643768206062270100098131812633714796206947998898813574041697679463706122684435120868673908214824392451997940519697
n+2 = 7621*10607*83911*p628
n+3 = 2*p641
n+4 = 3*19^2*23*43*67*419*433*557*5839*11005151*48791387*248743477*173166075757*176767927013058599371*3664337605725212245321703*38481802057600502530496251*263247353722581788826308137*10209898961843153162557563806797351795337*400438949478601771715861066925877874810624450405554075346315813*5481394579601493765715335676312985509068059008928260228225197198334163861140908690109*59671284938991446654929334084586000491614000771235438219780687791993969485740039897984101289374576781389292405323136206123396283987378378663819*3826571760000000347831791312832653173993441574924501382327329177624991886782939107192155367560772660694023438371884513433078262544237345403090052800073625471999
n+5 = 2^2*17*281*p637
n+6 = 5*7*71*3027473*p631
n+7 = 2*3*773*1699*10103853977418233687*41681046033582287059951*36621959455077044549010329*4211861143455373066167093525382847*86528790939635630219352122503699213*p498
n+8 = 5960137*45290243150024180117*p615
n-1 = 2*7*88309036301*7938470054221217579*c610
n+9 = 2^3*1138534864709243017*2353150702179440759951*28486552286765416493593*c578
n-1 and n+9 are unlikely to have other prime factors below 10^30.
There is a checked verification for k=9 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac6419.zip
k=9: 552 digits.
(n = (y-23^2)*(y-24^2)/55440-7, where y=(x*(5*x+9)/2-31)^2 and x=(1320*(10^20+13065906))^3)
n = 2^2*85580203*p544
n+1 = 3*p552a
n+2 = 2*7^2*16127*p546
n+3 = 17*19*29*47*107*2381*4817*4877*7243*8123*15349*351859*64106027*617354687*58939383403*2270969174255932789*64599558784499658763*70581857953182048626663359*832941909454872901874063981*38463978481445185081160367469*18565176607267555386412648342877*72712545267645351975447576546899*1872363765601959731971246441364247991*2945265097696419515747753341191750275309*316490425386673017172951555041414472713325780025891*5291973122549848056044955804501530796430017642474721*2451968078534651120722918940489036434974865520343721698595446659*27854698184101546744112650299911007030936127637904783943133412329
n+4 = 2^8*3^3*5^2*11^2*23^3*43^3*73^3*101*233*439*1579*1721*5641*5791^3*7001*25609^3*3059233*35685077*200804353^3*7372068659*5598060112993*23151817056913*38662729757040799*376176861511712141*4301348426994983857*15154667761484246406148307*1033141326087248524232877319*10617791649528225223660146631*391417105842348028109572403541777737773082843*4804384929120809533054017484874238511273037729623*1411936758329397253090018710058691271956742873966000085028776838783*831403915444995114442474420649091367670644872564910283524087531524227782803443372416623883166831227427855923684347833
n+5 = 223*9887*252181*p540
n+6 = 2*p552b
n+7 = 3*13*179*4651*4691*5419*14503*20483*45007*157291*3287266267*65755593703*22579302731919529*22650433627111273*254320536721426607*12212080029376746054763*174550854747787142801760373*404207064226822367082013951*452888967550655599403298024877*2271339091394769438948112369926152025853*336542104860088953870195699074182766905880245025434197664052101397215693118076537643469*507854227981676607382498641611332244205705154434984017560541283876746490947606161704323477*74631722951875434426641848605731026702747965564034236317050130044910555530463123439841102691867156169633878015853334887726181183
n+8 = 2^2*9479*90863*15503173813*p533
p552a and p552b are different 552-digit primes.
n-1 = 5*53*151*259499*c542
n+9 = 5*7*31*37*288232349*593280102498871*c524
n-1 and n+9 are unlikely to have other prime factors below 10^30.
There is a checked verification for k=9 in
http://physics.open.ac.uk/~dbroadhu/cert/ifacgg1.zip
k=9: 512 digits. Same as n+1 to n+9 for former 512-digit record for k=10.
k=9: 509 digits. Same as n+1 to n+9 for former 509-digit record for k=10.
k=10: 521 digits. Same as n+2 to n+11 in current record for k=12.
k=10: 515 digits. Same as n+1 to n+10 for former 515-digit record for k=11.
k=10: 512 digits.
(n = 4*(x-1)^2*(x^2+x+1)^2*(8*x^6-16*x^3+3)^2-9, where x=2*10^28+2204662)
n = 3^3*5*7*307*347*1187*9547*9697*15073*1531843*116110381*250277089*30578866463*111382910297*181158721739*446924830771*1867760574562610834993963*1488459549869389482650623003*8565397930808568846216431951262151*59769847830356822480851338650284237240697374480196117*300623037354997684760214480023188666593095587125093642757*20007803893063496839861142546375474591403195655644157207244713104487*3829797396593923020158058852535774081399736913920378739809259597238377868603562116552089648836044163530241297141741130983038474191260511793881126508518363309739
n+1 = 2^2*73*127*164881*1930288121*323256465041*90131193051180529*9303255826395894981169*5936198519764907325568788492837144034143764224325971823769414076715761951476514609129427797957422126713145651766928219379641251620723614216506369*3533742054174816074881641068866923118609347812218930248817864339645077320213576784897215431571348169712407642330812093231866486812662973436596195813720944828559128120676691802790771651955258619892556423839559184398326894169807919891967869197534212067400868464481337615037718331079392635959598394921
n+2 = 29*149*5901418121*p499a
n+3 = 2*3*499*2124821*13445407369*186826766724947388177434694180263630044871*95925605271645469122449575886666011652340523511711773270825209659498635468887476993851227773969941116968383660751*43778223112892451569864188936071609886439580867801305260814535751043759445135226017605038059205206222938075118226512125698742921604813012867568950280287564892654933848759437580955004560241577043651080237642470767694633379466907843523463211758931466512079866679115169383641194815890749469256483022750032129109031793983093916727712646977883
n+4 = 239*251*519846829*p499b
n+5 = 2^5*5*83*331*5413*1645601*13246099*831103267*442781686703*3271127680535279*6263540405958799*37333369881660386459*1730305848698194550273*34830720932848958310122536067257006458277*918030371532084486823140149211743092389268185858411703840428240548268686265325464850510544432796821911496795647262073608062725901351129418695163782977208778237112427886380213*8308098770575618563278140349708230566973798610696444827765518568173848312637469276391944462875073418410679255997789214467944099150573616414945633512750491792050045360588460558085423
n+6 = 3*11*13*23*41641*2894929*245823731*14734834751929*1248860046909467*1714352630976062017*68696646377151940356904991848691*826402995956294226375591690515963*17205979087403906335627860980115911179419243870279319985508971*4050624313071804187996566630261827216555059616139796924436663460527*12431506263961695057536834021915617377667459367414347205489562416699*1083419113812114581692806946928012991603417612837816992039548006438463*136522321437341357578982495960004506241249989563480223954886877392820440017871099119741773149006344599842089018411
n+7 = 2*7^2*17*31*191*449*3257*11068417*472514066396281571944993*81654641695766717900650891421256077306657*216432437109204897481725950423278756761076506433*207058905269150904115025146899823757889865604599393*103023928716175825862864924563274619099593728159866663736091990211978293021566631081305017858381298976256350246612433187999055255757621632033668070306853022509953*2359447004608294930877136571811981566820276927752364320206451612966434423077370369933866976913391759307946321794763605121319860302775451881015341532450655411517502262201
n+8 = 19*2927*5519*180473*1401203*11354621*18811567*113522593*14554641825673780279*179187138283689054518803*2100726463287606221888948966791*1235343327543615764457795495590772030447575628146102891785176718375548949498716110403276999113946325913624500215626384579131612707424323473497173390937204598816818971806525217501534898451*5268468875101207966503221637564275430828482281288806272352342013338174338145138429373064209988535633949696733331592548352583494113390219823457115218230511678697789109397011263873411779303893037684709359210406473
n+9 = 2^2*3^2*41^2*67^2*4099^2*31189^2*257987^2*2689200539563^2*233437103518022038867^2*741479532797940551572286509299641323^2*97584776774823127592117147231031959014393755760072393^2*8673383825406446118969387417345665128610749211198444515481^2*38250750512834751554907456243913073019849245258367375835003^2
p499a and p499b are different 499-digit primes.
n-1 = 2*1483*1933*21841*3791911*2358094721*71075504580403*c471
n+10= 5*37*272549*c505
1500 ECM curves at B1=1000000.
There is a checked verification for k=10 (also former record for k=9) in
http://immortaltheory.com/cnt/VERIFY_CONSEC.gp.
k=10: 509 digits.
(n = y*(y-23)*(y-41)*(y-64)/55440-4, where y=(13860*(10^60+1898683))^2)
n = 2^2*97*p506
n+1 = 3*19*443*p504
n+2 = 2*547*11551*2635904987582755423*57944354495692285367*p464
n+3 = 79*163*p505
n+4 = 2^4*3^2*5*7*11*17^2*29*199*569*3361*4783*6323*9871*27917*50666557*93548249*251419691*401234363*894724253*3501280801*4455055873*32445623759*156227602720306951640012557918423328611*206195408396375367587226324189737887279*50011980385626978791972075975276821553547719381*777768023292308549684490118627071144703463969328314641*58823529411764705882352941176470588235294117647058823641099*58823529411764705882352941176470588235294117647058823641099*17755515675727712860192365814762364249466181168372134315497429896352598545495965768332329930699664898435392329
n+5 = 31*37*47*653*919*125925257*7185033209071*112156733297849*328882642025827*79593348847085399*32883211741045463029*81115540365972458569*206921334578038811201*141676523493128094900799*1360783001639169584324399*58956746641484424911415529*233729786242778087790871756339*7748115078873569467096278095337513977*10185312414473538898820214907351318097219*113290015862499223221533450621707149727019427886981*821010301350627818810685153418641068867042983145299511812291*26557022341330289473110636903413510403866099214738127697823875132529407320405318271
n+6 = 2*167*152063*4590427*p495
n+7 = 3*89*112223*p501
n+8 = 2^2*13*16987*218249*356959*49526591*681273809*9106676377*19485931311178781*4967165135717508397*5499182353059262159*1001796858149748467827771*9069406043245243852023611*540526918608745881486206876437*2605643768471675941900168096886934928807*612167007455266691724205830229887098156127313*4307315168277185986198656054312024939833935058255871560498098605127723049*377554735891343947547657524494990782049555797943824283943584385821675669345829*171315516778112833179949573659386968855037125861456391334423927484245873649681869933101158002050587
n+9 = 5*880133*3463370921120207*232879941578044699*19041760972448845393261*p447
n was constructed such that n+4, n+5 and n+8 are each the product of known divisors with
at most 128 digits. Divisors of that size are possible to factor one at a time.
n-1 = 5^2*41*23689760674499*20201699120271441240691*c470
n+10= 2*3*23*449*304781745193811*47370240325563932561*432893534978100829771*c449
2000 ECM curves with B1=3000000, B2=4*10^9.
There is a checked verification for k=10 (includes former record for k=9) in
http://physics.open.ac.uk/~dbroadhu/cert/ifac10.zip.
k=11: 521 digits. Same as n+1 to n+11 in current record for k=12.
k=11: 515 digits.
(n = y*(2*y-11)^2/9-10 where y=(2*x^3-10*x-3)^2 and x=3*10^28+45140566)
n = 4253*226013*230357*416963*2861569*1880854277*42701426129693*13069902194569289*386588385678617711*500005063517754329373345969561796727584786706636720539573*4341248342043895913301460431960445634863440889511346793543879*47352224009928962954139286082311927636952488190447629980094200050881454389562654674388383721111298495341024455450378605701758635972436503172206041376444082433826157671195552288119771345608963121556060908695665231599547748300479567194952011445688867500985464164196239064860297609767831288273189825629275882499448373
n+1 = 2^5*3^3*5^2*17*647*7793*101027*387371*2876737*3190301*41234059*448017726073*4549647604301*7058560906391*82564644533297*313389057302287*547062257012137*176410881685979243*33529563487769597944543*3905797942598779457796869634480654828557063398268430209*5222735987089480599839721467537544683475254726525559707*126196121207067437991213318947836224809604713521480208688750619923880867*702704165961837523078979683266083025775067212012656945407515732734186659*184430385185788581648109351806679515278594015607579122998192486417070930340664716216441165690089873
n+2 = 49927*205863965163809*2931670092235410257*533262603377131368593*p456
n+3 = 2*7*19*29*31*197*12601*554284919*44696195304881*352313237452213*20936955960832581457*64810010594945926073*12486408345477232374233*823966587170283506206259822419*5493295331637810589063791182291419*290061485985760493441318238323975369*913349575170071565559047108472153379766319447*380514144459247521270215147347137364317548994601*39804603645056021960436070859321292255344545643899008580117*6911430939104636279124927560729205170078292892867255581576433921302778449907893288350968461824993673302019230391530510666265516371284599573288697014309779
n+4 = 3*97*173*p510a
n+5 = 2^2*3529*p510b
n+6 = 5*433*1511*5839*54877*70901*122219*304807*716819*1264213*1590521*2004001*6079961*602101170323*117556663309961*3809348476481813575538819198921*12831583161950832470071816910747*6431878224455115560889018029640943*4499242002482501219371685439637499957*1576088009746625511844899685544903239199*82167967255593167594806666269023272477390195236758647209681732983476457238690798651459261400520823548148134390254753813*17479824887805762459776111297664158300407464356342895406325546088030031811326503271187368529128996979561143882697282969022622722284955787
n+7 = 2*3*421*1931*p508
n+8 = 151*1067329*802239817*6599032263217*23817728203289561*883765947396190626787314222011033197735982012683064116394694384312468507385219425031573964810769850422257735631417704198370913509927691250034726865938966834569*613596283386608438258179573245611327009544063521439064418060929105150853857219744553356145561979397526887277403972646225035820934179866774672158662530913625446619207540431278244213112656505959897999772260600811020086509981580973246185253854296357393794483907554165864822538249281605736418530475133278726111833
n+9 = 2^3*89*101*113*179*521*8761*2996893*1385768831*4159235811161*21189480447750425779*746557470021770082838520077807*536425349409413320648023041422658535656199131337160825893487*4890845088036533014507974304316426334045685499764180221896202579301*3764607638136768105709541437596851092368933279685164279140523803173057942543926494529318439716150419*615012230835188011049059046392431382469745444689757743049199070972367899227064135417086557000328620017869735179939215758713862634224557943586663150054707229081347091467892171214006999591590114221
n+10= 3^2*7^2*11^4*13^2*929^2*4903^2*24497673032853345979^2*82548838381230577645194631472018766442910240852860847257^2*5284274390840591055900162350171249633036536528107034354024855679025659783826864691^2*65289483804720539669247608448854856335421158107620593110415396059725296456406814441995691^2
p510a and p510b are different 510-digit primes.
n-1 = 2*11*2377*1861437640330348699939070011*483058876581817566371212578148357*c450
n+11= 2*5*18289*5453317*4322671468897*105475130567783166422952459096726461*c455
n-1 and n+11 are unlikely to have other prime factors below 10^35.
There is a checked verification for k=11 in
http://physics.open.ac.uk/~dbroadhu/cert/ifac515g.zip
Made by Jens Kruse Andersen,
jens.k.a@get2net.dk home
First version 6 August 2007. Last updated 25 December 2018.