Date: Mar 19, 2000 4:45 AM
Author: Milo Gardner
Subject: Unitary (Egyptian) fractions

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.

Modern number theory suggestions that only look at algorithms, and

not the historical algebraic identity methodsm such as:

1. n/p - 1/A = (nA -p)/Ap

as the 300 BC Hibeh n/45 table generalized in the composite for,

2. n/pq = 1/A + (NA -pq)/Apq

and the older composite form,

n/pq = n/A x A/pq

where A = 5, 25 in the EMLR

A = (p + 1), (p + q) in the RMP 2/nth table

and other methods in later Greek texts like the Akhmim Papyrus

(and its interesting n/17 and n/19 table, for example) have

interest Kevin Brown.

I wonder if others have taken the modern algorithmetic methods and

considered testing their 'logic' by trying to create small last

terms series, as Fibonacci and the modern view of the 1202 Liber

Abaci has so often been stated?

As another idea, why not try the historical diophantine indeterminate

equations actually used by Fibonacci himself, as Heing Lueneburg

(U. of Kasierslatern) listed on historia matematica a few months

ago? You will find that Leonard Pisani used two indeterminate

equation methods to solve for n/p and n/pq, as revised from the

much older algebraic identity methods that I cited above. For

example, for the selection of highly compoiste A's in n/p series, that

the RMP, chosen from the range p/n < A < p, Fibonacci

extended his serch to 2p.

Clearly the introduction of base 10 decimals and its algorithmic

view of number has blinded many from reading the ancient texts

in the simplicity, as briefly outlined above.

Regards to all,