Breaking the RMP 2/nth table code

Thank you for considering this solution to the 1650 BC Rhind Mathematical Papyrus 2/nth table. A number theory method has been applied. This previously unsolved number theory from ancient Egypt, from 2,000 BC covers a period of over 3,000 years (before base 10 decimals) wrote fractions as noted by 2/nth table rule as used in the 500AD Akhmim Papyrus, the 1650 BC Rhind Papyrys and 2,000 BC Moscow Papyrus.

The Akhmim Papyrus, a Hellene 500 AD to 800 AD papyrus, found along the Nile, was cited cited by Wilbur Knorr, Stanford History of Science Department, in Historia Mathematica, HM 9, "Fraction in Ancient Egypt and Greece". Knorr's excellent paper included several tables such as n/17 and n/19 that connect to 1650 BC Egypt by:

 
 
                         One set of decodings of late Hellene
n/17   Akhmim P. Value   views of 1650 BC Egyptian Fractions
----  ----------------  -------------------------------------
2/17   12' 51' 68'          2a - p = 7 [a = 12, divisors 4, 3]
3/17   12' 17' 51' 68'      2/17 + 1/17
4/17   12' 15'17' 68' 85'   (1/3 + 1/17)(1/4 + 1/5) or
                            (1/a + 1/p)(1/u + 1/v)*
4/17   12' 15' 17' 68' 85'  (1/3 + 1/17)(1/4 + 1/5) or
            3/17 + 1/17     (1/a + 1/p)(1/u + 1/v) + 1/17
5/17   4' 34' 68'           5a - p [a = 10, divisors 2, 1]
6/17   3' 51'               6a - p = 1 [ new general form]
7/19   3' 17' 51'           6/17 + 1/17
8/17   3' 15' 17' 85'       (1/3 + 1/17)(1/1 + 1/5)
9/17  1/2 34'               5/17 + 5/17 - 1/17
10/17 1/2 17' 34'           5/17 + 5/17
11/17 1/2 12' 34' 51' 68'  10/17 + 2/17 (12' 51' 68')
12/17 1/2 12' 17' 34' 51' 68'  11/17 + 1/17
13/17 1/2  4' 68'           9/17 + 5/17 - 1/17
14/17 1/2  4' 17' 68'      13/17 + 1/17
15/17 1/2  3' 34' 51'       9/17 + 6/17
16/17 1/2  3' 17' 34' 51'  15/17 + 1/17

 
                             On set of decoding of Hellene
n/19    Akhmim P. value      views of 1650 BC Egyptian fractions
-----  --------------------- ------------------------------------
 2/19  10' 190'               2a- p = 1 is one of three methods
 3/19  15'  20' 57' 76' 95'   (1/3 + 1/4 + 1/5)(1/1 + 1/19) - 2/3**
                           or 2/19 + 2/19 - 1/19 [a = 30, divisors 6, 5]
 4/19   5' 95'                2/19 + 2/19
 5/19   4' 76'                2a - p [ one of three methods]
 6/19   4' 19' 76'            5/19 + 1/19
 7/19   3' 38' 114'           7a - p = 2 [ a = 3, divisors 2, 1]
                           or (1/2 + 1/6)(1/1 + 1/19) - 2/3**
 8/19   3' 30' 38' 57' 95'    (1/2 + 1/3 + 1/5)(1/1 + 1/19) - 2/3**
 9/19   3' 12' 38' 57' 76'    (1/2 + 1/3 + 1/4)(1/1 + 1/19) - 2/3**
10/19  1/2 38'                5/19 + 5/17
11/19  1/2 19' 38'            10/19 + 1/19
12/19  1/2 12' 38' 76' 114'   11/19 + 1/19 with new 5/19 + 5/19
                              [5/19 = 12' 76' 114', a = 12, divisors 3, 2]
13/19   3" 57'                 3" = 2/3 and 7/19 + 7/19 - 1/19
14/19   3" 19' 57'            13/19 + 1/19
15/19  1/2  4' 38' 76'        10/19 + 5/19
16/19  1/2  4' 19' 38' 76'    15/19 + 1/19
17/19  1/2  3' 30' 57' 95'    18/19 - 1/19
18/19  1/2  3' 12' 57' 76'    1/2  + 9/19 - 1/38

The Akhmim Paprus' full use of p/q as an elementary number theory element is easily coinnect to 2000 BC - 1650 BC Egyptian fractions by the 2/nth beginning terms.

In one manner of speaking I solved this problem amateur mathematician. As the esteemed history of mathematics journal HISTORIA MATHEMATICA reviewed in its Feb. 1995 issue, HM 22, Sylvia Couchoud's 208 page amateur paper was worthy of a closer look by professional mathematical historians.

Maurice Caveing's review of Sylvia Couchoud's work is totally in French, I may have been able to read about 75% of Maurice's comments, but even on that level I found a review of 2,000 BC Egyptian fractions has been grossly under valued by Egyptologists and math historians.

You may be able to appreciate the following facts:

1. Prior to 2,000 BC Egyptian fractions followed a binary structure, with the notation being called Horus-Eye, as noted by

   1 = 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + ... + 1/2n + ...

2. Babylonian base 60 followed a very similar structure such that zero was not required to be used. Only the fractions needed were listed. No zero place holders were required, as our base 10 decimal system required.

3. By 2,000 BC Babylonian algebra has been reported by the majority of mathematical historians, such as Boyer in his popular text, that this rhetorical algebra is equivalent to our modern algebra I.

4. Yet even the algebraic geometry listed in the Moscow Papyrus, 2,000 BC, as noted by Couchoud, continues to under valued, as connected to an unworthy form of Egyptian hieratic arithmetic.

5. By 1850 BC, as recorded in 1650 BC by Ahmes in the Rhind Mathematical Papyrus a hieratic form of fractions alters greatly from the earlier Horus -Eye hieroglyphic fraction.

6. History of Science authors like Neugebauer, Gillings and Knorr have cited a consistent composite number pattern, as I prefer to write as:

   2/pq = (1/q + 1/pq)2/(p + 1) where p and q are prime with p > q.

7. Neugebauer notes the general algorithmic aspect of the composite form, as does Gillings for the multiple of 3 case, and as does Knorr.

8. Disagreement between scholars is noted on two levels for the exceptions 2/35, 2/91 and 2/95.
   1. First is the three composite case exceptions 2/35 and 2/91.

      Knorr discusses the 2/35 and 2/91 in a friendly manner, while Neugebauer and Gillings were not as kind.

      Concerning 2/35 and 2/91 a well known pattern does emerge, refuting the conclusions set down by scholars, a form that is clearly an inverted Greek Golden Proportion, the product of the arithmetic mean and the harmonic mean.

      Note that the arithmetic mean A = (p + q)/2 and the Harmonic mean H = 2pq/(p + q) can be seen as

      2/AH = 2/pq = (1/p + 1/q)2/(p + q).

      Fill in the values for p = 5, q = 7 for 2/35 and p = 7 and q = 13 for 2/91 and see what I mean.

      As a 500 AD to 800 AD Akhmim Papyrus point, the Egyptian inverted Golden Proportion seems to be improved upon, as Howard Eves noted in his AN INTRODUCTION TO THE HISTORY OF MATHEMATICS textbook, by:

      z/pq = 1/pr + 1/qr where r = (p + q)/2

   2. Second is the exception 2/95, which is really:

      2/19 stated as a prime unit fractions time 1/5.

      Here the prime unit fraction algorithm is revealed by:

      2/p = 1/a + (2a - p)/ap where

      a is a highly divisible number, about 2/3rds the value of p.

      Using 2/19 as an example a = 12 was chosen by Ahmes such that of the two sets of divisors of a that add to 5 < 2a - p, (2(12) - 19) = 5> 4,1 and 3,2 to be specific the largest smallest term seemed to appeal to Ahmes.

      Writing out

      2/19 = 1/12 + (3 + 2)/(12*19)
      
           = 1/12 + 1/76 + 1/114 or
      	

      as Greeks wrote,

      2/19 = 12' 76' 114'

      It should be noted at the prime number pattern point that two aspects are significant. First, and most importantly, all prime numbers in the RMP 2/nth table follow this rule. Since there are no exceptions could it be that the famous Sieve of Eratosthenes was anticipated by over 1,500 years? Second, the essentials of the aliquot part algorithm, divisors of the first partition, was noted by B.L. van der Waerden in SCIENCE AWAKENING, about 30 years ago.

I hope you find the first cut of this Egyptian fraction interesting. Two 15 page papers are available that detail these newly decoded historical points in greater depth, such as discussing Boyer when he concluded 2,000 BC Egyptians did not consider p/q as an elementary element (within a domain of natural numbers).

Sincerely,

Milo Rea Gardner
Cryptanalyst and
Math Historian

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