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Topic: unitary (Egyptian) fractions
Replies: 17   Last Post: Mar 31, 2000 6:38 AM

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 Milo Gardner Posts: 1,105 Registered: 12/3/04
Unitary (Egyptian) fractions
Posted: Mar 19, 2000 4:45 AM

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,