I found an easy way to generate those sets, and to prove that they are complete.
For the "starting from two digits" version, neither one of the exposed sets for 7 and 8 are complete. Some larger primes are still lurking in the dark there. I have the complete sets for both 8, and 7 for the both cases when the base itself is included in the set or not*, but I don't want to spoil the puzzle, this is an interesting little problem... hehe...
Hint:
Code:
gp > a=(7^17-5)/2
%1 = 116315256993601
gp > isprime(a)
%2 = 1
gp > digits(a,7)
%3 = [3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 3, 1]
gp >
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*when the base is prime, like for 5 and 7, the sets are different; including the base results in automatic elimination of all possible extension numbers with "0 after 1" from the set, which is quite restrictive, so I also calculated the lists for the "base is not included" version, i.e. base-5 starting from 6, and base-7 starting from 8; in this case, for example, base-5 will include numbers like 104 and 10103 which are prime, and base-7 list will include 1022, 1051, 1202, .... 1100021 ... etc, they are "enriched" compared with the case when the first "10" is included. So I have the complete list for 8, and the complete two lists for 7, the normal one, and the "enriched" one. Base-5 is easy, in any case.