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:

    2/n = 2/n*(m/m)
        = 2m/mn
        = (mn1 + ... + mni)/(mn)
        = 1/a + 1/a1(n) + ...+ 1/ai(n)

where:

    1. 2m was additively partitioned into i integers: (mn1 + mn2 + ... +
       mni) = 2m, based on mn being factored into mn1, mn2, ..., mni and
       other primes and composites

such that:

    2. mn1, mn2, ..., mni each divide mn

and,

    3. m = a, in most cases, when for m > 8

and,

    4. As an observation a1, ..., ai were always divisors of the
       denominator a of the first partition 1/a

example:

    2/7 = 2/7*(4/4) = 8/28 = (7 + 1)/28 = 1/4 + 1/28

where:

    2m = 8, with additive parts (7 + 1) = 8

and,

    a = 4, with 4 and n: being divisors of 28 (and all a1, a2, ... denominators).

The second-step, 2m/mn (8/28) has been omitted. The omission provides a
sense of Ahmes' red auxiliary shorthand, a major decoding barrier. The
decoded data therefore leaves work for readers, as Ahmes did himself, to
compute 2m and confirm the numerator's additive parts that adds to 2m.

RMP 2/n TABLE: DECODED

    2/3   = 1/3 + 1/3 = 2/3 followed an EMLR 1/3 = 1/6 + 1/6 'rule'.
    2/5   = 2/5*(3/3) = (5 + 1)/15 = 1/3 + 1/15
    2/7   = 2/7*(4/4) = (7 + 1)/28 = 1/4 + 1/28
    2/9   = 2/9*(2/2) = (3 + 1)/18 = 1/6 + 1/18
    2/11  = 2/11*(6/6) = (11 + 1)/66 = 1/6 + 1/66
    2/13  = 2/13*(8/8) = (13 + 2 + 1)/104 = 1/8 + 1/52 + 1/104
    2/15  = 2/15*(2/2) = (3 + 1)/30 = 1/10 + 1/30
    2/17  = 2/17*(12/12) = (17 + 4 + 3)/204 = 1/12 + 1/51 + 1/68
    2/19  = 2/19*(12/12) = (19 + 3 + 2)/228 = 1/12 + 1/76 + 1/114
    2/21  = 2/21*(2/2) = (3 + 1)/42 = 1/14 + 1/42
    2/23  = 2/23*(12/12) =(23 +1)/276 = 1/12 1/276
    2/25  = 2/25*(3/3) = (5 + 1)/75 = 1/15 + 1/75
    2/27  = 2/27*(2/2) = (3 + 1)/54 = 1/18 + 1/54
    2/29  = 2/29*(24/24)= (29 + 12 + 4 + 3)/696 = 1/24 + 1/58 + 1/174 + 1/232
    2/31  = 2/31*(20/20) = (31 + 5 + 4)/1620 = 1/20 + 1/124 + 1/155
    2/33  = 2/33*(2/2) = (3 + 1)/66 = 1/22 + 1/66
    2/35  = 2/35*(30/30) = (35 + 25)/1050 = 1/30 + 1/42
    2/37  = 2/37*(24/24) = ( 37 + 8 + 3 )/888 = 1/24 + 1/111 + 1/296
    2/39  = 2/39*(2/2)= (3 + 1)/78 = 1/26 + 1/78
    2/41  = 2/41*(24/24)= (41 + 4 + 3 )/984 = 1/24 + 1/246 + 1/328
    2/43  = 2/43*(42/42)=(43 + 21 + 14 + 6)/1806 = 1/42 + 1/86 + 1/129 + 1/301
    2/45  = 2/45*(2/2)      = ( 3 + 1)/90 = 1/30 + 1/90
    2/47  = 2/47*(30/30)    = (47 + 10 + 3)/1410 = 1/30 + 1/141 + 1/470
    2/49  = 2/49*(4/4)      = (7 + 1)/196 = 1/28 + 1/196
    2/51  = 2/51*(2/2)      = (3 + 1)/102 = 1/34 + 1/102
    2/53  = 2/53*(30/30)    = (53 + 5 + 2 )/1590 = 1/30 + 1/318 + 1/795
    2/55  = 2/55(6/6)       = (11 + 1)/330 = 1/30 + 1/330
    2/57  = 2/57*(2/2)      = (3 + 1)/114 = 1/38 + 1/114
    2/59  = 2/59*(36/36)    = (59 + 9 + 4) /2124 = 1/36 + 1/236 + 1/531
    2/61  = 2/61*(40/40)    = (61 + 10 + 5 + 4)/2440 = 1/40 + 244 + 1/488 + 1/610
    2/63  = 2/63*(2/2)      = (3 + 1)/126 = 1/42 + 1/126
    2/65  = 2/65*(3/3)      = (5 + 1)/195 = 1/39 + 1/195
    2/67  = 2/67*(40/40)    = (67 + 8 +5 )/2680 = 1/40 + 1/335 + 1/536
    2/69  = 2/69*(2/2)      = (3 + 1)/138 = 1/46 +1/138
    2/71  = 2/71*(40/40)    = (71+ 5 + 4)2840 = 1/40 + 1/568 + 1/710
    2/73  = 2/73*(60/60)    = (73 + 20 + 15 + 12)/4380 = 1/60 + 1/219 + 1/292 + 1/365
    2/75  = 2/75*(2/2)      = (3 +1)/150 = 1/50 + 1/75
    2/77  = 2/77*(4/4)      = (7 + 1)/388 = 1/44 + 1/308
    2/79  = 2/79*(60/60)    = (79 + 20 + 15 + 6 )/4740 = 1/60 + 237 + 1/316 + 1/790
    2/81  = 2/81*(2/2)      = (3 + 1)/162 = 1/54 + 1/162
    2/83  = 2/83*(60/60)    = (83+ 15 + 12 +10)/4980 = 1/60 + 1/332 + 1/415 + 1/498
    2/85  = 2/85*(3/3)      = (5 + 1)/255 = 1/51 + 1/255
    2/87  = 2/87*(2/2)      = (3 + 1)/174 = 1/58 + 1/74
    2/89  = 2/89*(60/60)    = (89 + 15 +10 + 6)/5340 = 1/60 + 1/356 + 1/534 + 1/890
    2/91  = 2/91*(70/70)    = (91 + 49)/6370 = 1/70 + 1/130
    2/93  = 2/93*(2/2)      = (3 + 1)/186 = 1/62 + 1/186
    2/95  = 2/95*(12/12)    = (19 + 3 + 2)/1140 = 1/60 + 1/380 + 1/570
    2/97  = 2/97*(56/56)    = (97+ 8 + 7 )/5432 = 1/56 + 1/679 + 1/776
    2/99  = 2/99*(2/2)      = (3 + 1)/198 = 1/66 + 1/198
    2/101 = 2/101*(6/6)     = (6 + 3 + 2 + 1)/606 = 1/101 + 1/202 + 1/303 + 1/606

OBSERVATIONS

1. When n is a prime > 99

a. The EMLR cited: m = 6 was used to convert 1/101, as Ahmes used m = 6 to convert 2/101 and likely all larger 2/p prime denominators.

b. By implication (since Ahmes did not mention this class of detail) when n was composite, m was selected in terms of its largest prime number. Taking two examples, n = 19 and n = 95 (5*19), Ahmes used m = 12.

2. When m > 8

a. mn1 = n

b. a = m

3. The 2/n table method was extended to n/p and n/pq tables. A Coptic Akhmim Papyrus
included over-analyzed n/17 and n/19 data, referenced in the INTRODUCTION. Applying Occam's Razor to simplify Coptic multiples several examples vulgar fractions may be of interest:

a. 2/19 x 10/10 = (19 +1)/190 = 10' 190' (Greek notation 1/n = n'):
b. m = 60 to convert 3/19, additive parts (76 + 57 + 20 + 15 + 12)/1140
c. m = 6 to convert 4/19,
d. m = 4 to convert 5/19.
e. m = 6 to convert 6/19
f. m= 6 to convert 7/19
g. m = 30 to convert 8/19

CONCLUSION

Summing up Ahmes' stated, and observed, RMP 2/n table rules, a three-step
method included the use of intuition. Ahmes also understood aspects of the
fundamental theorem of arithmetic, that integers uniquely factor into prime
numbers. It should be noted that Ahmes did not use algorithms, though
fragments of a single false position algorithm have been reported for
almost 100 years, and a cursive algorithm was known in the Old Kingdom.
Middle Kingdom single false position was not used in RMP 31, and other like
problems, since traditional red auxiliary multiples offers a simpler
approach. In RMP 31 Ahmes converted 28/97 by solving two vuglar fractions:
26/97 + 2/97. Ahmes solved 26/97 (with m = 4) and solved 2/97 (with m =
56). Ahmes combined 28/97 unit fractions series into one answer.
Robins-Shute' RMP 1987 book discusses the problem/answer, omitting the red
auxiliary step, a common practice of historians. Historians need to
included a red auxiliary step even in texts that scribes left no explicit
clue of its use. The Egyptian fraction data provides sufficient proof of
the scribal use of multiples, every time when 2/n, and vulgar fractions
were optimally or elegantly created.

Concerning a wider view of vulgar fractions and 2/n tables a narrative
links Old Kingdom infinite series numeration to Middle Kingdom finite
Egyptian fraction numeration. Ahmes' red auxiliary first-step, used in 2/n
tables, was extended to convert n/p and n/pq vulgar fractions solved in 84
RMP problems. Ahmes, RMP 81, discussed two additional classes of Egyptian
fraction series. The additional classes defined Egyptian fraction weights
and measures applications that partitioned a hekat unity (64/64), writing
red auxiliary remainders, and m/n 'sub-units' of a hekat, that did not use
red auxiliary remainders.

BIBLIOGRAPHY

1. Mahmoud Ezzamel, Accounting for Private Estates and the Household in the
   20th Century BC Middle Kingdom, Abacus Vol 38 pp 235-263, 2002

2. Milo Gardner, The Egyptian Mathematical Leather Roll Attested Short Term
   and Long Term, History of Mathematical Sciences, Hindustan Book Company,
   2002.

3. Milo Gardner, An Ancient Egyptian Problem and its Innovative Solution,
   Ganita Bharati, MD Publications Pvt Ltd, 2006.

4. Richard Gillings, Mathematics in the Time of the Pharaohs, Dover Books,
   1992.

5. Heinz Lüneburg, Leonardi Pisani Liber Abbaci oder Lesevergnügen eines
   Mathematikers, Mannheim: B. I. Wissenschaftsverlag , 1993.

6. Oystein Ore, Number Theory and its History, McGraw-Hill, 1948.

7. T.E. Peet, Arithmetic in the Middle Kingdom, Journal Egyptian
   Archeology, 1923.

8. Tanja Pommerening, "Altagyptische Holmasse Metrologish neu
   Interpretiert" and relevant phramaceutical and medical knowledge, an
   abstract, Phillips-Universtat, Marburg, 8-11-2004, taken from "Die
   Altagyptschen Hohlmass, Buske-Verlag, 2005.

9. L.E. Sigler, Fibonacci's Liber Abaci: Leonardo Pisano's Book of
   Calculation, Springer, 2002.

10. Hana Vymazalova, The Wooden Tablets from Cairo:The Use of the Grain
    Unit HK3T in Ancient Egypt, Archiv Orientalai, Charles U Prague, 2002.

LINKS:

    1. RMP 2/n Table (Wikipedia)
    2. EMLR (Wikipedia)
    3. EMLR (Planetmath)
    4. Hultsch-Bruins Method (Planetmath)
    5. Egyptian fractions (Planetmath)
    6. Kahun Papyrus
    7. Liber Abaci (Planetmath)
    8. Liber Abaci (Blog)

