```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 thatthe first term of the series can be 2/3 or 3/4. Considering that732/733 is requires a large first term, I suspect that one DavidEppstein's 40 + algorithmic methods, or one of Kevin Brown 'inversemethods' should create a series shorter or equal to the Horus-Eye5-term series. YES, TO BE EGYPTIAN 5-TERM OR SHORTER SERIES is arequest when converting any rational number to a finite series.Modern number theory suggestions that only look at algorithms, andnot the historical algebraic identity methodsm such as:1. n/p - 1/A = (nA -p)/Apas the 300 BC Hibeh n/45 table generalized in the composite for,2.  n/pq = 1/A + (NA -pq)/Apqand 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) haveinterest Kevin Brown.I wonder if others have taken the modern algorithmetic methods andconsidered testing their 'logic' by trying to create small lastterms series, as Fibonacci and the modern view of the 1202 LiberAbaci has so often been stated? As another idea, why not try the historical diophantine indeterminateequations actually used by Fibonacci himself, as Heing Lueneburg(U. of Kasierslatern) listed on historia matematica a few monthsago? You will find that Leonard Pisani used two indeterminateequation methods to solve for n/p and n/pq, as revised from the much older algebraic identity methods that I cited above. Forexample, for the selection of highly compoiste A's in n/p series, thatthe RMP, chosen from the range p/n < A < p, Fibonacciextended his serch to 2p.Clearly the introduction of base 10 decimals and its algorithmicview of number has blinded many from reading the ancient textsin the simplicity, as briefly outlined above.Regards to all,
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