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http://rmprectotable.blogspot.com/2008/07/rmp-2n-table.html

BREAKING the RMP 2/n Table Code
Author: Milo Gardner Bio

INTRODUCTION

Breaking the RMP 2/n table code is a central task for historians that wish to
learn Egyptian mathematics. Egyptians wrote rational numbers in two ways. The
first method acted like our modern rational numbers. The second method,
written in a unique manner, converted rational numbers to equivalent unit
fraction, sometimes optimized and sometimes not, to concise series.

The first, and second methods, are recorded in the Kahun Papyrus and the Rhind
Mathematical Papyrus. Both texts solved arithmetic progression problems by
using rational number differences. Egyptian fraction series were used to write
final answers. Arithmetic progressions, an algebra topic that followed
interesting formulas, will not be discussed. The second method, an arithmetic
topic, 2/n tables, listing optimized unit fraction series, will be discussed
in terms of a likely ancient unifying theme.

Overly analytical views of possible 2/n table methods, especially potential
2/35, 2/91 and 2/95 conversion methods have been suggested by scholars for
over 100 years. In 1995, an interesting internet paper suggested discussed the
2/35, 2/91 and 2/95 cases, as well as 2/17 and 2/19 table solutions written in
500 AD (Akhmim Papyrus). Overall, modern number theory's abstract analysis are
important. These analysis show that ancient Egyptian fractions were likely
well definedm and unified in several respects. But, what where the ancient
ideas that unified the work of Ahmes and other ancient scribes? Can the
ancient unifying idea(s) be identified?

As a method to attempt to correct for over analytical and under analytical
errors, and fairly parse ancient unifying ideas, Fibonacci's 1202 AD Liber
Abaci was consulted. The first 126 pages of the Liber Abaci covers
conversion and factoring examples. The examples are summarized by seven
Egyptian fraction conversion methods. Four of the conversion methods date
to the time of Ahmes. It will be shown that Ahmes used a three-step
conversion method that applied four of Fibonacci's seven conversion
methods. Ahmes first step, the selection of a multiple, was indirectly
cited in Ahmes' shorthand notation by a' red auxiliary number. Hence the
selection of particular multiples may have been intuitive.

CODE BREAKING METHODOLOGY

It will be shown that Egyptian fraction series were created from vulgar
fractions by using optimal red auxiliary numbers for over 3,000 years. The
Egyptian fraction body of knowledge was reverse and forward engineered.
Historians parse aspects of the oldest rational number arithmetic by
working backwards. Historians consider original and additional Egyptian
fraction patterns by analytical methods. The test for historians is to find
the simplest forward engineered method. The traditional scholarly method
points out original methods by applying Occam's Razor. In the case of
Ahmes' RMP 2/n table , Occam's Razor was applied to point out a general
used of red auxiliary multiple. A 350 year older text, the EMLR details 26
conversions of 1/p and 1/pq by selecting non-optimal multiples.

Following a Middle Kingdom tradition, texts published in the Ancient Near
East for 3,000 years published optimal and elegant Egyptian fraction texts.
Ahmes' tradition optimized red auxiliary numbers. The multiple step will be
denoted by a number m. For ease of reporting 2/n table data, m will be
replaced by (m/m). First-steps, second-steps, and third-steps were
personalized by scribes. That is, Ahmes' 2/n table rules report alternative
multiples, when compared to other Egyptian fraction texts. For example
alternative multiples are reported in the Kahun 2/n table. Alternative
multiples are also reported in Greek, Ancient Near East, and medieval
texts.

Whatever set of conversion steps Ahmes may have used, Ahmes, and other
scribes, first converted vulgar fractions by multiplying 2/n vulgar
fractions by red auxiliary numbers (m/m). Two additional steps were used by
Ahmes when 2/n table information was fully translated into modern
arithmetic. Clearly Ahmes' arithmetic partitioned numerator 2m into
additive integers, a fact pointed out in the 1920's. That is, 2/n
conversions created a new numerator 2m, a new denominator mn, a set of
additive 2m integers, and an optimized Egyptian fraction series. Ahmes
method is symbolically written as Rule One, a fact decoded in 2005.

Rule One: