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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?

Andrés E. Caicedo
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tomerg
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1 Answers1

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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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    For those that cannot follow the link, it essentially says that a greedy algorithm works: simply keep subtracting the largest reciprocal not already used which is less than the residual, and eventually this will terminate because, once you stop using the initial terms of the harmonic series, the numerator of the residual will strictly reduce with each step until it reaches zero. – Henry May 15 '11 at 21:41
  • @tomerg: Please accept an answer so that this question is not in the unanswered list. –  May 17 '11 at 06:38
  • Jstor link for that article: http://www.jstor.org/stable/2315211 – Martin Sleziak Nov 27 '11 at 12:17