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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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    Do you want ALL the digits to be used, or possibly just some of them?2012-06-23
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    @tomasz: all the given digits.2012-06-23
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    You probably want $1, because e.g., $333\ldots 333$ is always a multiple of $3$.2012-06-23
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    @Aaron: any number written only with $0,3,6,9$ is a multiple of $3$, much the same as with $0,2,4,8$ and $2$, or just $a>1$ and $a$ itself. But I think there are infinitely many primes of the form $1111\ldots1$. In any case, it does not really depend on $k$, rather, on the digits picked.2012-06-23
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    @tomasz Yes, but I'm reading the problem as "given any subset of the decimal digits, are the infinitely many primes built only using those digits" and I'm saying that if the subset has size one, then all but one of those subsets fail for trivial reasons. If I am reading the problem wrong, that is another issue.2012-06-23
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    @Aaron: I'm suggesting that it is not the right question, the right one being: for what sets of digits are there infinitely many primes written using only those digits. Also, if $1,0$ are among the chosen digits, then the answer is YES.2012-06-23
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    There are many sets that allow at most one-digit primes: any subset of $\{0,2,4,5,6,8\}$, any subset of $\{0,3,6,9\}$, and any singleton except $\{1\}$. It is not known whether there are infinitely many primes using only the digit $1$, though it is conjectured that this is the case. As @tomasz says, a better question would be which subsets allow the construction of infinitely many primes, though in at least some cases the answer simply isn’t known.2012-06-23
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    Now, it has been modified to a more specific question!2012-06-23
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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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