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        riddles >> medium >> Fractions
        
(Message started by: BenVitale on Feb 18^th, 2008, 9:32am)

Title: Fractions
Post by BenVitale on Feb 18^th, 2008, 9:32am

A unit fraction is a rational number with numerator 1. Is it true that every positive rational number can be written as the sum of distinct positive unit fractions? If so, how do you do it; if not, why not?

Title: Re: Fractions
Post by Eigenray on Feb 18^th, 2008, 11:27am

The greedy algorithm [hide]does work. Try to find a pattern[/hide].

Title: Re: Fractions
Post by BenVitale on Feb 18^th, 2008, 12:43pm

I searched and found a link that helps

http://kevingong.com/Math/EgyptianFractions.pdf

Title: Re: Fractions
Post by cool_joh on Feb 18^th, 2008, 7:05pm

This (http://mathnerds.com/mathnerds/best/MagnificentSeven/solution.aspx) may help.

Title: Re: Fractions
Post by Eigenray on Feb 18^th, 2008, 7:50pm

When applying the greedy algorithm to x, the first n terms are H_n = 1+1/2+1/3+...+1/n, the largest harmonic number < x. After this, we always take the largest unit fraction less than the remainder. Once we reach this point, what happens to [hide]the numerator of the remainder[/hide]?