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?