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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.2012-07-06
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    Simple heuristic arguments point to "no", but I think this problem is unsolved. See also the "86 conjecture" http://math.stackexchange.com/questions/25660/status-of-a-conjecture-about-powers-of-22012-07-06
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    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.2012-07-06

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