I suspect in that in any base b, and for any collection of base b digits such that the gcd of the whole set is 1, there are infinitely many primes which only us

I suspect in that in any base b, and for any collection of base b digits such that the gcd of the whole set is 1, there are infinitely many primes which only use those digits. I also suspect the Mersenne case b=2, set={1}, is the hardest.