
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 HorusEye 5term series. YES, TO BE EGYPTIAN 5TERM 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,

