The Egyptian Mathematical Leather Roll (EMLR) dates to the 1850 BC to 2000 BCE period of the Egyptian Middle Kingdom. The leather scroll has been housed in the British Museum from 1864 to the present. The leather roll was not softened and unrolled until 1927. Scholars. initially had correctly reported the the text's practical arithmetical relationships, but overlooked theoretical considerations.

The text is an important historical document since it defines small rational numbers, 1/p and 1/pq, that scribes converted to Egyptian fraction series. Modern scholars had failed to consider scribal uses of p, and q, as prime numbers until the 1990's. Today, the EMLR is seen as converting 22 1/p, and 1/pq fractions to Egyptian fraction series using one or two basic methods.

There were six classes of 1/p, and 1/pq conversions. The first five classes multiplied the rational number by a multiple of 2, 3, 4, 5, (or 6?), 7, or 25. Each conversion obtained not-so-elegant Egyptian fraction series answers.

Concerning theoretical details, the EMLR student was asked to find multiples of several unit fractions, one being 1/8. The 1/8 rational number was scaled to three multiples, one 3/3, written as:

In total, the EMLR, converted 22 rational numbers using six multiples: 2/2, 3/3, 4/4, 5/5, 7/7, and 25/25, and one identity:

or a multiple of 6, per:

1/101 * 6/6 = 6/606 = (3 + 2 + 1)/606 = 1/202 + 1/303 + 1/606

Six classes outline 22 rational numbers, and 26 answers. The first multiple, 2/2, did not calculate a standard Egyptian fraction series, writing: 1/3 = 1/6 + 1/6.

The six EMLR classes of answers were deciphered in algebraic form in the 1990's. A closely related arithmetic multiple method was noted in the Liber Abaci. The Liber Abaci method apparently was 3,000 years older, possibly originating in the EMLR. The Liber Abaci method(s) consisted of working through the EMLR's patterns by asking: what was the simplest method that the scribe could have used? A common link connected the EMLR data points to a multiple of each unit fraction. The simplest method, decoded in the EMLR, also appeared in the Liber Abaci, suggested by Siger (the 2002 translator of the Latin text) by using the phrase: first distinction.

The entire RMP 2/n table can be read as using either the EMLR multiple method, or the generalized Liber Abaci multiple method. For example, the RMP's 2/pq conversions were raised to a multiple

The RMP 2/pq multiple method found all but three of its Egyptian fraction series, with 2/35, 2/91 and 2/95 being the exceptions.

Returning to the EMLR, seven categories (A -G) summarize the use of multiples of 2, 3, 4, 5, 7, 25, plus one identity, 1 = 1/2 + 1/3 + 1/6.

A. Four examples: 1/5, 1/3, 1/2, and 2/3 were converted unit fraction series. Each unit fraction was decomposed into repeating unit fractions by using a multiple of 2 or a multiple of 3 of the initial unit fraction. The raw data follows with [ ]being added to complete the logical calculation.

1.

2.

3.

4.

The EMLR introduces the student to the easiest rational numbers, 1/p and 1/pq, converting the them Egyptian fraction using two basic methods, a multiple and an identity.

The EMLR converted 22 rational numbers, with the student using either a multiple of 3, 4, 5, 7 or 25 or an identity method. Answers were written in not-so-elegant Egyptian fraction series. The most common of the 22 rational number problems was: find an Egyptian fraction series based on scaling the initial unit fraction to a multiple of itself. The EMLR text list five classes of multiples: multiple of 3, 4, 5, 7, 25, and one identity:

A summary of the five multiple classes, and one identity case: (B -G) follows:

B. Multiple of 3 (10 or 11 questions)

There were 11 rational numbers in which the student was asked to use a multiple of 3, restating the initial unit fraction by 1/n x 3/3, finding 3/3n, or using another method. The student created not-so- elegant Egyptian fraction series by finding least common denominators (LCMs) such that: 1/6, 1/8, 1/10, 1/12, 1/14, 1/16, 1/20, 1/30, 1/32 and 1/64 were converted by the following steps (as implied by the EMLR student's answers):

5.

6.

7.

8.

9.

10.

11.

12.

13.

14.

and most likely,

15.

The 1/13 problem contains an error, 3/49. The student may have been asked to write out the answer as a multiple of 3, a multiple of 7 problem, or by some other method. Gillings had supposed that may have facilitated the appropriate conversion method. Whatever the proper question, and expected conversion method, the student mistakingly converted 3/49 obtaining an incorrect Egyptian fraction series. The student may have intended to begin with 3/39, a multiple of 3. Whatever the error the EMLR shows that 1/13 was incorrectly converted to:

Working the incorrect answer backwards, looking for another basis of the student's error, the problem can be seen as a multiple of 7 problem, or:

an answer that the student's mentor may have expected.

Stated as a multiple of 3, or multiple of 7. question, the student may have not been expected by his mentor to answer either question. That is, the mentor may have not have expected the student to solve 2/13, or 7/13, rational numbers that the student was unprepared to solve.

It should be noted that the RMP 2/nth table and its set of of 2/p and 2/pq methods offered methods to solve n/p and n/pq problems.

But, I am getting ahead of the discussion. Returning to multiples beyond the initial set of 10 or 11 multiple of 3 questions, the student scribe correctly wrote answers to the following questions.

C. Multiple of 4 (1 question) 1/4

16.

D. Multiple of 5 (3 questions): 1/4, 1/5 and 1/15

17.

18.

19. 1/15 = [(1/5)*(5/15)

as the scribe had restated 5/15 as 10/30, allowing:

with 10 being written as

10 = 6 + 3 + 1, such that

E. Multiple of 7 + 1/4

20 1/4 = [(1/7)* (7/4) = (4 + 2 + 1)/4

F. Multiple of 25 (2 questions) 1/8, 1/16

21. 1/8 = [25/400 = (1/5)*(3/5 + 1/40)

22. 1/16 = [(1/2)*(1/25 + 1/15 + 1/75 + 1/200)]

G. Identity statements (4 questions) 1/7, 1/9, 1/11, 1/15

23. 1/7 = [(1/7)*(1/2 + 1/3 + 1/6)]

24. 1/9 = [(1/9)*(1/2 + 1/3 + 1/6)]

25 1/11 = [(1/11)* (1/2 + 1/3 = 1/6)]

26. 1/15 = [(1/15)*(1/2 + 1/3 + 1/6)]

In summary, the EMLR was an answer sheet. It was used in a course that taught its student to convert rational numbers to Egyptian fraction series. Beyond the EMLR's 1/p, and 1/pq answers, advanced methods were required to convert 2/p, and 2/pq, to Egyptian fraction series. One advanced method, the Hultsch-Bruins (H-B) method, may have been a multiple method, 6/6 in the EMLR, and modified to consider the RMP 2/p first partitions as a multiple. The formal H-B method converted 2/p rational numbers. It was apparently unknown to the EMLR student. This is, the EMLR incorrectly converted 1/13. Had H-B been available, several solutions may have been available.

Finally, advanced students, studying beyond the EMLR text, converted n/p and n/pq to quotients and Egyptian fraction remainders. Several large vulgar fractions like 1554/97, and 1386/97 were solved in RMP algebra problems. Advanced scribes converted almost any rational number to an integer quotient, and an exact Egyptian fraction remainder. In conclusion, scribes employed a small set of proto-number theory tools. The tools allowed scribes to generally convert n/p and n/pq to Egyptian fractions, with the EMLR text detailing the simplest methods.