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My main question is the generalization, though one can answer the first one and it will get accepted.

  • Are there infinitely many primes involving $3,7$ only?

Generalization: For what sets of given $k$ distinct digits (not all even) from $\{0,1,...,9\}$ where $1\leq k \leq 9,$ there are infinitely many prime numbers involving only these $k$ digits?

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    An interesting consequence if the predicate is false for the set $\{1, \dots, 9\}$ is that *all* primes greater than$a$certain maximum would contain a $0$.2017-06-25

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I note that Primes that contain digits 3 and 7 only are tabulated at the Online Encyclopedia of Integer Sequences. My understanding is that there is no set $D$ of fewer than 10 digits for which it has been proved that there are infinitely many primes which use only the digits in $D$.