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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?

  • 2
    So, [Egyptian fractions](http://mathworld.wolfram.com/EgyptianFraction.html)?2011-05-15
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    You might want to say **positive** rational numbers since negative rationals obviously cannot be formed with numerator 1 and natural-number denominators. (The Putnam problem cited by Chandru1 specifies that the numbers be positive.)2011-05-15
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    @Fixee, natural numbers are positive.2011-05-15
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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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