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Finding a CPAP-k with large difference is harder because more numbers must be composite at the same time. The table shows the single largest known CPAP-k difference (in bold) for each k. If more than one CPAP is known with that difference then only the first found is shown. Only the minimal difference k# = 210 has been found for CPAP-8, -9 and -10.
The largest known CPAP-k difference
k Primes n's Digits Year Discoverer(s)
3 x2506 + 21102n n=0..2 2506 2004 Torbjörn Alm & Jens Kruse Andersen, PrimeForm, Primo
4 25900 + 469721931951 + 2880n n=0..3 1777 2007 Ken Davis, NewPGen, PrimeForm, Primo
5 9400734826 · 1499# + x632 + 2310n n=0..4 645 2004 Jim Fougeron, Primo
6 x218 + 840n n=0..5 218 2004 Torbjörn Alm & Jens Kruse Andersen
7 x133 + 420n n=0..6 133 2004 Torbjörn Alm & Jens Kruse Andersen
8 x92 + 210n n=0..7 92 1997 Harvey Dubner, Tony Forbes & Paul Zimmermann
9 500996388736659 · 193# + x76 + 210n n=0..8 92 1998 Manfred Toplic, CP09
10 507618446770482 · 193# + x77 + 210n n=0..9

93

 1998 Manfred Toplic, CP10

The following table shows current and old records, some from before the record category was added here. The first discovered CPAP-7, -8, -9 and -10 are all listed. I would guess the first CPAP-7 was also the first CPAP-5 and -6 with difference 210, and that no larger difference was known until the listed records.
Hans Rosenthal's A probable CPAP-3 with d=17676 is not listed because no primality proofs were performed.
History of the largest known CPAP-k difference
k Primes n's Digits Year Discoverer(s)
3 x2506 + 21102n n=0..2 2506 2004 Torbjörn Alm & Jens Kruse Andersen, PrimeForm, Primo
3 41#137 - 5576107 + 10164n n=0..2

1985

 2002 David Broadhurst, Primo
 
4 25900 + 469721931951 + 2880n n=0..3 1777 2007 Ken Davis, NewPGen, PrimeForm, Primo
4 46313478 · 1201# / 1302643 + x498 + 2310n n=0..3 505 2004 Jens Kruse Andersen, PrimeForm, Primo
4 78006074 · 883# + x371 + 2004n n=0..3 379 2004 Jim Fougeron
4 23320 + 1308319536235 + 1470n n=0..3 1000 2004 Hans Rosenthal, Jim Fougeron, Primo
 
5 9400734826 · 1499# + x632 + 2310n n=0..4 645 2004 Jim Fougeron, Primo
5 x272 + 1350n n=0..4 272 2004 Jens Kruse Andersen
 
6 x218 + 840n n=0..5 218 2004 Torbjörn Alm & Jens Kruse Andersen
 
7 x133 + 420n n=0..6 133 2004 Torbjörn Alm & Jens Kruse Andersen
7 x97 + 210n n=0..6 97 1995 Harvey Dubner & Harry Nelson
 
8 x92 + 210n n=0..7 92 1997 Harvey Dubner, Tony Forbes & Paul Zimmermann
 
9 500996388736659 · 193# + x76 + 210n n=0..8 92 1998 Manfred Toplic, CP09
 
10 507618446770482 · 193# + x77 + 210n n=0..9

93

 1998 Manfred Toplic, CP10