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### Fibonacci Algorithm and Egyptian Fractions

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Date: 04/25/99 at 19:06:37
From: liz
Subject: Egyptian fractions

I would like to know how to do 4 rounds in the Fibonacci algorithm. I
have been able to get 3 rounds (521/1050 = 1/3 + 1/7 + 150 ) - could
you please show me how to do 4 rounds with 2 examples? Also, I would
like to know whether it is always possible to change any fraction into
an Egyptian fraction using the decomposition method.

Thanks.
Liz
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Date: 04/26/99 at 16:30:42
From: Doctor Rick
Subject: Re: Egyptian fractions

Hi, Liz, welcome to Ask Dr. Math!

The number of terms in an Egyptian fraction found by the Fibonacci
fractions that lead to 4 or more terms using this method; one of these
(the one with the smallest denominator) is 8/11. Try 16/17 too.

You can easily write an Egyptian fraction as long as you want. For
instance, any unit fraction 1/n in your expansion can be replaced by

1/n = 1/(2n) + 1/(3n) + 1/(6n)

521/1050 = 1/3 + 1/7 + 1/50
= (1/6 + 1/9 + 1/18) +
(1/14 + 1/21 + 1/42) +
(1/100 + 1/150 + 1/300)

There, we just multiplied the number of terms by 3, and you can
multiply it by 3 again by replacing each of these terms, and so on
forever. So finding long Egyptian fractions isn't a challenge; but
finding the SHORTEST Egyptian fraction expansion for any given
fraction is a challenge. The Fibonacci algorithm isn't guaranteed to
find the shortest.

Take a look at this interesting Web site:

An Introduction to Egyptian Fractions (Ron Knott)
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fractions/egyptian.html

It has lots of interesting information about Egyptian fractions,
including the outline of a proof that the Fibonacci algorithm will
produce an Egyptian fraction of finite length for any fraction. I
don't know what you mean by the "decomposition method" so I can't say
whether it will always work, if it's different from the Fibonacci
method (and there are lots of methods).

- Doctor Rick, The Math Forum
http://mathforum.org/dr.math/
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Associated Topics:
High School Fibonacci Sequence/Golden Ratio

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