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The minimal CPAP-k
k Primes n's Digits Year Discoverer(s)
3 3 + 2n 0..2 1    
4 251 + 6n 0..3 3    
5 9843019 + 30n 0..4 7    
6 121174811 + 30n 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 0..6 23
 2018 Paul Zimmermann
7(2)382003672700092872707633 + 210n0..624
2018Paul Zimmermann
7(3) 2210835776623037377907953 + 210n 0..6 25
 2018 Paul Zimmermann
8(1) 2799806429564 • 83# • 113 + x34 + 210n 0..7 47 2004 Hans Rosenthal & Jens Kruse Andersen
8(2) 5351738881202 • 83# • 113 + x34 + 210n 0..7 48 2004 Hans Rosenthal & Jens Kruse Andersen
8(3) 16003606986539 • 83# • 113 + x34 + 210n 0..7 48 2004 Hans Rosenthal & Jens Kruse Andersen
9(1) 3416716311814 • 179# / (149 • 157) + x65 + 210n 0..8 79 2004 Hans Rosenthal & Jens Kruse Andersen
9(2) 12606057030290 • 179# / (149 • 157) + x65 + 210n 0..8 80 2005 Hans Rosenthal & Jens Kruse Andersen
9(3) 52515434335080 • 179# / (149 • 157) + x65 + 210n 0..8 80 2005 Hans Rosenthal & Jens Kruse Andersen
10(1) 507618446770482 • 193# + x77 + 210n 0..9

93

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

93

 2008 Manfred Toplic, CP10