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Is it true that every positive rational number $r = \frac{n}{m}$ could be represented as sum of $\frac{1}{k}$ for different $k$'s:

$$ r = \sum_{i=1}^{s} \frac{1}{c_i} $$ where all $c_i$ are different?

If true, are there any bounds on number of summands $s$ based on $n$ and $m$?

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closed as no longer relevant by Gjergji Zaimi, Robin Chapman, Victor Protsak, falagar, Pete L. Clark Jul 23 '10 at 10:40

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google for "egyptian fractions". –  Franz Lemmermeyer Jul 22 '10 at 13:56
4  
It's a nice exercise to prove this for yourself. In fact (hint), you can show that the greedy algorithm always works here: at each stage, choose $\frac{1}{c_i}$ to be as large as possible so that the partial sum is less than or equal to $r$. –  Pete L. Clark Jul 22 '10 at 14:03
    
yes, en.wikipedia.org/wiki/Egyptian_fraction has a complete exposition of the problem. –  Wadim Zudilin Jul 22 '10 at 14:30
    
I think that the question about bounds on $s$ is still meaningful, but the reference Vose (1985) in the WP article seems to address it. –  Victor Protsak Jul 22 '10 at 21:17
    
There are currently some votes to close. The commenters have mentioned the existence of a wikipedia article which at least addresses all of the OP's questions. If the OP has further or more specific questions which are not completely answered by that article, it would be a good idea to edit the question accordingly before it gets closed. –  Pete L. Clark Jul 23 '10 at 6:04

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