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Can it be shown that a positive fraction of the base-$b$ digits of n! are nonzero (in the limit as $n\to\infty$)?

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The answer: $9/10$ of course but even $> 0$ is most possibly unprovable presently. I asked a similar question about powers of an integer (say, $3$) here: mathoverflow.net/questions/38971/… (see Update 4 in that question). –  Mark Sapir Dec 24 '11 at 0:10
I meant $(b-1)/b$, of course. –  Mark Sapir Dec 25 '11 at 14:08

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up vote 12 down vote accepted

I believe the best current lower bound on this is the one given by F. Luca in "The Number of Non-Zero Digits of n!" Canad. Math. Bull. 45(2002), 115-118. It is proven there that the number of non-zero base $b$ digits grows at least as fast as $C_b\log n$.

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Interesting! It is better than the current lower bound for $a^n$ (where $a$ is co-prime with 10). –  Mark Sapir Dec 24 '11 at 0:39

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