Date: Mar 20, 2000 3:25 PM
Author: eppstein@euclid.ics.uci.edu
Subject: Re: Unitary (Egyptian) fractions



milogardner@juno.com (Milo Gardner) writes:
> It is interesting that no one has taken the Egyptian position that
> the first term of the series can be 2/3 or 3/4. Considering that
> 732/733 is requires a large first term, I suspect that one David
> Eppstein's 40 + algorithmic methods, or one of Kevin Brown 'inverse
> methods' should create a series shorter or equal to the Horus-Eye
> 5-term series. YES, TO BE EGYPTIAN 5-TERM OR SHORTER SERIES is a
> request when converting any rational number to a finite series.


I don't think I really have 40 different methods, and they're mostly not
original to me.

Using 2/3 at least would be more historically accurate if less
aesthetically pleasing. However there is no 5-term series for 732/733
using 2/3, it requires six, e.g.

732/733 = 2/3 + 1/5 + 1/8 + 1/145 + 1/17592 + 1/63771
= 3/4 + 1/5 + 1/24 + 1/145 + 1/17592 + 1/63771

(The requirement for six terms follows easily from the facts that
732/733 needs seven unit fractions and that 2/3 or 3/4 can be expanded
into two unit fractions each.)
--
David Eppstein UC Irvine Dept. of Information & Computer Science
eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/