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Problem: Prove that all positive rational numbers can be expressed as the finite sum of different numbers $\displaystyle \frac {1} {n}$ ($n$ is a natural number).

Example: $\displaystyle \frac {19}{16}=1+ \frac {1}{8} + \frac {1}{16}.$

*We cant sum numbers as $\displaystyle \frac {3}{16}$ (denominator > 1) but we can sum $\displaystyle \frac {1}{8}+ \frac {1}{16}.$

Any solutions? Suggestions?

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    @quanta: Exactly. Which is why it's unlikely you can form a negative **rational** number via a sum of fractions with 1 over a **natural** number.2011-05-15

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This is a putnam problem. For a complete solution please look here.

  • Take a look at this article as well: J.C.Owings, American Mathematical Monthly Vol. 75 (1968), Pages $777-778$.
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    Jstor link for that article: http://www.jstor.org/stable/23152112011-11-27