Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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

share|improve this question
1  
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
add comment

1 Answer

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 $C_b\log n$.

share|improve this answer
    
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
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.