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The minimal CPAP-k
k Primes n's Digits Year Discoverer(s)
3 3 + 2n n=0..2 1    
4 251 + 6n n=0..3 3    
5 9843019 + 30n n=0..4 7    
6 121174811 + 30n n=0..5 9 1967 L. J. Lander & T. R. Parkin

The minimal CPAP-k is currently only known for k<7.
After that the prime difference must be at least 210 and the minimal solution is probably so large that an exhaustive search for it would be extremely hard.
Heuristics (estimates based on probability) indicate the minimal CPAP-7 may have around 21 digits. As of November 2018, the smallest known has 23 digits.


The need for at least 209 6 = 1254 composites in a CPAP-k with k>6 means it is much harder to find CPAP's with small primes than larger ones.
The below table shows the 3 smallest known CPAP-k when the minimal is unknown. There are only two known CPAP-10.

The smallest known CPAP-k
k Primes n's Digits Year Discoverer(s)
7(1) 71137654873189893604531 + 210n n=0..6 23
2018 Paul Zimmermann
7(2)382003672700092872707633 + 210nn=0..624
2018Paul Zimmermann
7(3) 2210835776623037377907953 + 210n n=0..6 25
2018 Paul Zimmermann
8(1) 2799806429564 83#113 + x34 + 210n n=0..7 47 2004 Hans Rosenthal & Jens Kruse Andersen
8(2) 5351738881202 83#113 + x34 + 210n n=0..7 48 2004 Hans Rosenthal & Jens Kruse Andersen
8(3) 16003606986539 83#113 + x34 + 210n n=0..7 48 2004 Hans Rosenthal & Jens Kruse Andersen
9(1) 3416716311814 179# / (149157) + x65 + 210n n=0..8 79 2004 Hans Rosenthal & Jens Kruse Andersen
9(2) 12606057030290 179# / (149157) + x65 + 210n n=0..8 80 2005 Hans Rosenthal & Jens Kruse Andersen
9(3) 52515434335080 179# / (149157) + x65 + 210n n=0..8 80 2005 Hans Rosenthal & Jens Kruse Andersen
10(1) 507618446770482 193# + x77 + 210n n=0..9

93

1998 Manfred Toplic, CP10
10(2) 1180477472752474 193# + x77 + 210n n=0..9

93

2008 Manfred Toplic, CP10