# Largest Consecutive Factorizations

Introduction
News
Records
Former records
Record lengths
Factorizations
Factorizations for former records
Credited programs and projects
External links

### Introduction

This page lists the largest known case of k consecutive numbers for which the complete prime factorization is known. Only records with at least 500 digits are listed. This currently gives records for k = 2 to 13. Prime factorization is usually much harder than primality testing. It is unrealistic to factor a random number above 500 digits if the second largest prime factor has more than 100 digits. But if a lot of examples are tried then cases where the second largest factors are small may be found. There are also algebraic methods to ensure relatively small prime factors in some of the consecutive numbers.

### News

2018
December 7. 24862048-digit record for k = 2 by Patrick Laroche and GIMPS.

2017
December 26. 23249425-digit record for k = 2 by Jonathan Pace and GIMPS.
May 31. 2971-digit record for k = 6 by Oscar Östlin.
May 24. 12202-digit record for k = 4 by Anand Nair, Oscar Östlin.

2016
March 19. 3414-digit record for k=5 by Oscar Östlin, Primo.
March 13. 1493-digit record for k=7 by Oscar Östlin, Primo.
February 29. 388342-digit record for k = 3 by Scott Brown, PrimeGrid, TwinGen, LLR.
January 7. 22338618-digit record for k = 2 by Curtis Cooper and GIMPS.

2015
November 24. A second 2158-digit record for k = 5 by Oscar Östlin.
October 20. 2158-digit record for k = 5 by Oscar Östlin.
February 8. -digit record (with a prp factor) for k = 4 by Oscar Östlin.

2013
January 25. 17425170-digit record for k = 2 by Curtis Cooper and GIMPS.

2012
April 11. 200701-digit record for k = 3 by Philipp Bliedung, PrimeGrid.
February 29. 10673-digit record for k = 4 by Tom Wu.
February. 2155-digit record for k = 5 by Dirk Augustin.

2011
December 25. 200700-digit record for k = 3 by Timothy D. Winslow, PrimeGrid.
December 25 (found in 2010 but not published until it was beaten by the above). 100355-digit record for k = 3 (with a prp factor) by Jens Kruse Andersen using a prime found by Mark Simpson.

2010
May 3. 8193-digit record for k = 4 (with proven prime factors) by Tom Wu and Geoffrey Hird.
April 28. 10673-digit record for k = 4 (with a prp factor) by Tom Wu.
January 24. 5638-digit record for k = 4 (with proven prime factors) by Tom Wu.

2009
December 22. 850-digit record for k = 8 by David Broadhurst.
December 20. 5257-digit record for k = 4 (with proven prime factors) by Tom Wu.
December 19. 2139-digit record for k = 6 (also record for k=5) by David Broadhurst.
December 16. 500-digit record for k = 13 by Joe Crump and John Michael Crump. The first case of k = 13.
December 7. 552-digit record for k = 11 (also record for k=10) by Joe Crump and John Michael Crump.
December 6. 804-digit record for k = 9 (also record for k=8) by Joe Crump and John Michael Crump.
December 6. 521-digit record for k = 12 (also record for k=10 and k=11) by David Broadhurst. The first case of k = 12.
November 29. 515-digit record for k = 11 (also record for k=10) by David Broadhurst. The first case of k = 11.
November 23. 641-digit record for k = 9 by David Broadhurst.
(November 17. A 641-digit case for k = 8 is announced by David Broadhurst shortly after the below 703-digit record by Joe Crump, so the 641-digit case was not a record and is not listed on this page.)
November 17. 703-digit record for k = 8 by Joe Crump and John Michael Crump.
November 11. 552-digit record for k = 9 by David Broadhurst.
November 7. 512-digit record for k = 10 (also record for k=9) by Joe Crump and John Michael Crump.
August 13. 100355-digit record for k = 3 by (SG Grid), Peter Kaiser, Keith Klahn, Twin Prime Search, PrimeGrid.

2008
October 9. 4187-digit record for k = 4 (with proven prime factors) by Matthew Peets and Jens Kruse Andersen using a CC2 found by Markus Frind.
September 16. 12978189-digit record for k = 2 by Edson Smith and GIMPS.

2007
September 28. 10043-digit record for k = 4 (with a prp factor) by Jens Kruse Andersen using a prime found by Ken Davis.
September 20. 74595-digit record for k = 3 (with a prp factor) by Jens Kruse Andersen using a prime found by Tony Galvan.
September 19. 74288-digit record for k = 3 (with a prp factor) by Jens Kruse Andersen using a prime found by Peter Benson.
September 17. 67218-digit record for k = 3 (with a prp factor) by Jens Kruse Andersen using a prime found by Thomas Ritschel.
September 16. 64868-digit record for k = 3 (with a prp factor) by Jens Kruse Andersen using a prime found by Chris Chatfield.
September 14. 60222-digit record for k = 3 (with a prp factor) by Jens Kruse Andersen using a prime found by Jiong Sun.
September 7. 1803-digit record for k = 6 by David Broadhurst.
September 2. 600-digit record for k = 8 by David Broadhurst.
August 29. 3500-digit record for k = 4 (with proven prime factors) by Donovan Johnson using primes found by Gary Barnes.
August 27. 6223-digit record for k = 4 (with a prp factor) by Christophe Clavier using primes found by Norman Luhn.
August 27. 1404-digit record for k = 7 (also record for k=6) by David Broadhurst.
August 20. 1404-digit record for k = 6 by David Broadhurst.
August 18. 2135-digit record for k = 5 by Donovan Johnson.
August 14. 1104-digit record for k = 6 by David Broadhurst.
August 11. 509-digit record for k = 10 (also record for k=8 and k=9) by David Broadhurst.
August 9. 509-digit record for k = 8 by David Broadhurst.
August 6. This page opens with newly set records for k = 4, 6, 7 by Jens Kruse Andersen using primes found by Markus Frind and Paul Underwood.

### Records

The factorizations are in a later section linked in the k column. The prime factorization of a prime is the prime itself, so a known prime has known prime factorization. Some of the records were not found by starting from scratch with the purpose of setting this type of record. Instead, they use one or more primes from some "origin" to reduce the amount of numbers that must be factored. If the largest known case with at least one prp (probable prime) factor is above the largest with proven factors then both are shown. A year link is to an announcement. Please mail new records you find or know about. Include prime factorizations and used programs.
The largest known k consecutive factorizations
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

Former records for largest known k consecutive factorizations
(improved since this page opened 6 August 2007)
k n (first number) Digits Origin Year Discoverers
2 277232917 − 1 23249425 Mersenne prime 2017 Jonathan Pace, GIMPS
2274207281 − 122338618Mersenne prime2016Curtis 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

Record lengths
First known case of k consecutive factorizations with at least 500 digits.
See Jaroslaw Wroblewski's Longest Consecutive Factorizations for record lengths above 300 digits.
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

### Factorizations

The prime factorizations for the records are given below. If n is composite then the above table expression is given in parentheses. Some large prime factors are given as pN with N an integer. At most one factor per number is given like this so the value can be computed by dividing the original number by the other prime factors. pN is an N-digit prime. prpN is an N-digit probable prime.
cN is an N-digit unfactored composite number that prevents a record extension to a larger k. The amount of ECM factoring I am aware of on cN is given. It's with GMP-ECM and default B2 when B2 is not given. Other factoring work like P-1 and P+1 is not listed.
```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.
```

### Factorizations for former records

Some of these might become new records for a larger k value if a composite factor in n-1 or n+k is factored. However, there is yet no record published on this page which has later been extended to larger k.
```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
```

### Credited programs and projects

Programs:
NewPGen by Paul Jobling.
APSieve by Michael Bell and Jim Fougeron.
tpsieve by Geoff Reynolds.
TwinGen by David Underbakke.
PrimeForm by the OpenPFGW group with George Woltman.
Proth.exe by Yves Gallot.
LLR by Jean Penné.
Prime95 by George Woltman.
Primo (formerly Titanix) by Marcel Martin.
GMP-ECM by Paul Zimmermann, Alexander Kruppa and others.
PARI/GP by Henri Cohen, Karim Belabas and others.
Msieve by Jason Papadopoulos.
GGNFS by Chris Monico.

Projects:
GIMPS by George Woltman, Scott Kurowski, et al.
Twin Prime Search and PrimeGrid, coordinated by Michael Kwok, Andrea Pacini, Rytis Slatkevicius.
Primesearch organized by Michael Hartley.

### External links

Jack Brennen, Consecutive numbers all factored. 2002 post which helped inspire this page.
David Broadhurst, A Chinese Prouhet–Tarry–Escott solution (pdf). Work inspired by this record page.
Jaroslaw Wroblewski, Longest Consecutive Factorizations. The longest sequence above 300 digits.
Yahoo! Groups, primeform. Mailing list which includes discussion of consecutive factorizations.
mersenneforum.org, Factoring Projects. General factoring forum.

Made by Jens Kruse Andersen, jens.k.a@get2net.dk   home
First version 6 August 2007. Last updated 25 December 2018.