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Are there infinitely many $$n\in\mathbb N$$ such that $3^n$ has all digits non-zero?

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No. Of course not. But I cannot prove this (correct) answer. – GEdgar Jul 6 '12 at 20:35
Simple heuristic arguments point to "no", but I think this problem is unsolved. See also the "86 conjecture"… – Cocopuffs Jul 6 '12 at 20:55
Hendrik Lenstra says that recreational number theory is "that branch of Number Theory which is too difficult for serious study." I think you have a good example of that here. – MJD Jul 6 '12 at 23:10

The known powers of 3 with no zeros are tabulated here. There are 22 of them, the largest being $3^{68}$. There is a link there to a page where it is claimed that the search of numbers $3^k$ has gone out to $k=10^8$ without finding any more. So I think GEdgar has it about right in his comment.

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