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Is there a positive number $n$ of distinct odd integers $z_1,z_2, \ldots, z_n \geq 3$ such that $\frac{1}{z_1} + \frac{1}{z_2} + \cdots + \frac{1}{z_n} = 1$?

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A related question is whether 2 can be the sum of the inverses of the positive divisors of an odd integer. But this is an open problem because it is logical equivalent to the open question whether there is an odd perfect number. –  Steffen Schuler Jan 13 '13 at 6:33

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up vote 7 down vote accepted

In 1954, it was shown by Stewart and Breusch (independently) that if $\frac {p}{q} >0$ and $q$ is odd, then it can be written as the sum of finitely many reciprocals of odd numbers.

As a specific example,

$$1=\frac {1}{3} + \frac {1}{5} + \frac {1}{7} + \frac {1}{9} + \frac {1}{15} + \frac {1}{21} + \frac {1}{27} + \frac {1}{35} + \frac {1}{63} + \frac {1}{105} + \frac {1}{135}$$

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A similar question, are there infinitely many such nontrivial representations? By what you said, we can continually decompose each of the above fractions, but we will always get that the sum of finitely many of them will be, say $\frac{1}{3}$. Is there a representation that doesn't use $1/3$ (or any particular fraction)? –  Clayton Jan 13 '13 at 4:49
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@Clayton You can certainly repeat this procedure infinitely often with the smallest fraction. There is a lower bound on the number of representations with an odd number of terms, and I think it grows exponentially. (There is clearly no way to do it with an even number of terms). –  Calvin Lin Jan 13 '13 at 4:53
    
@SteffenSchuler That's a very different question though. –  Calvin Lin Jan 13 '13 at 6:35
    
@CalvinLin: They are indeed different but also related, because if we would suppose in an indirect proof that there were an odd perfect number, we can reduce this assumption to my main question. If that would be solved negatively (it is now solved positively), we would have proven that there is no odd perfect number. –  Steffen Schuler Jan 13 '13 at 15:38

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