INTRODUCTION: The Rhind Mathematical Papyrus, written by Ahmes in 1650 BCE, and the Kahun Papyrus, written 200 years earlier, both began with 2/n tables. Scholars have debated the topic for 130 years with little success during the 19th and 20th centuries. How did Middle Kingdom scribes generally convert 2/p, 2/n, n/p and n/pq to optimized, but not optimal, unit fraction series?

The first number theory solution, using aliquot parts, to the 2/p problem was suggested by F. Hultsch in 1895. Hultsch considered even denominators' first partitions that were available between p/2 and p. Each denominator was inspected for composite and prime divisors. For example, to convert 2/19 the denominators of possible first partitions: 1/10, 1/12, 1/14, 1/16, and 1/18 were considered as the aliquot parts of 12 (12, 6, 4, 3, 2 and 1) such that 2/19 - 1/12 created a remainder 5/(12*19). The remainder's numerator 5 = (3 + 2) was additively obtained from the set 12, 6, 4, 3, 2, 1, in this case 5= (4 + 1). Ahmes optimized aliquot part selections of divisors denominators, by first selecting a LCM multiple of 2/n to calculate additive numerators of final unit fraction series.

Ahmes' LCM method considered aliquot parts of the divisor of the first partition by selecting a LCMs with additive aliquot parts written in red in numerators. RMP 36 describes details of the LCM and aliquot method. Composite and primes factor first partition denominators optimized the conversion of n/p and n/pq by picking a LCM that created additive red auxiliary numbers.

Before discussing the medieval multiple method, let's look at the 2/p Egyptian fraction series via Hultsch-Bruins. The first RMP 3-term series is 2/19. The rational number 2/19 was converted to an optimal Egyptian fraction series by considering alternative first partitions, even denominators between 19/2 and 19. Alternative first partitions are subtracted from the fraction from 2/19. Ahmes was assisted by an LCM method marked in red in the RMP to find the optimal first partition. There are five possible first partitions: 1/10, 1/12, 1/14, 1/16, and 1/18. Ahmes selected 1/12 without an detailed explanation. However, by inserting aliquot parts, decoding patterns were exposed by Hultsch. Ahmes converted 2/19 by considering the divisors of 12 (12, 6, 4, 3, 2, 1) by a mental process. A scribal shorthand set red auxiliary numbers were applied such that 2/19 = 1/12 + 5/(12*19), was the mental process, and $$2/19*(12/12)= 24/228 = (19 + 3 + 2)/228 = 1/12 + 1/75 + 1/114$$ , as the formal method.

Ahmes converted the numerator 5 be found by selecting two or more divisors of 12. In descending order unit fraction series were written. Ahmes considered two solutions (4 + 1) and (3 + 2). Ahmes selected (3 + 2) by writing 2/19 = 1/12 + (3+2)/(12*19) = 1/12 + 1/76 1/114 in the ancient Egyptian fraction notation. Note that nearly optimal first partitions and nearly optimal red number multiples solved 2/n conversion problems.

Considering 2/91, often noted by historians as an odd Egyptian fraction series, its solution is obtained by Hultsch-Bruins. Ahmes selected the first partition 1/70 after considering several even first partitions between 1/46 and 1/90. Ahmes inspected the denominator of 1/70, and found composite and prime divisors: 35, 14, 7, 5, 2, 1, such that 49, taken from the remainder (140 - 91)/(70*91), found 35 + 14 allowing:

2/91 = 1/70 + 49/(70*91) = 1/70 + 1/130

by the shorthand method

and,

2/91*(70/70) = 140/6370 = (91 + 49)/6340 = 1/70 + 1/130

by the formal multiple method.

As a medieval alternative, that had been passed down though Egyptian, Greek, Hellene, and Arab cultures, Ahmes could have selected single multiples that converted all 2/n table members. Yes, a single method. The single multiple method also appears in 24 of 26 series of the 200 year older Egyptian Mathematical Leather Roll. The two exceptions used two multiples depicted by an out-of-order series, capturing multiples 25 and 6 being applied.

In 1944, E. M. Bruins independently verified Hultsch's 49 year old solution to the 2/$n$ th table problem. Math historians and Egyptologists had debated the validity of the Hultsch-Bruin method for over 60 years. A long overdue honor to F. Hultsch needs to be published.

OPTIMAL MULTIPLE INFORMATION: In 2002 a complete Latin to English translation of Leonardo de Pisa (Fibonacci)'s Liber Abaci became available. Fibonacci's well known book was translated by L.E. Sigler. Fibonacci had practiced a 3,200 year old number theory craft by primarily considering a multiple method. Fibonacci also used a second partition method. When first partitions did not convert to an Egyptian fraction series second partitions solved the problem. For code breakers the Liber Abaci changed the RMP 2/n table debate. The Liber Abaci showed that the Hultsch-Bruins method was a shorthand method that found optimized, but not optimal, RMP 2/n table Egyptian fraction series. Ahmes discussed several details of his shorthand in RMP 36 and the 2/n table.

The Liber Abaci summarizes three versions of the Hultsch-Bruins method. Yet, it too used multiples in the style of Ahmes. Sigler's seventh section describes seven medieval and ancient Egyptian fraction methods. Methods four and five discuss Leonardo examples that may detail a H-B method. Multiple methods dominated Fibonacci's Egyptian fraction writings. Leonardo selected unit fraction first partitions, subtracting it from the vulgar fraction being converted to an elegant Egyptian fraction series. Method six included a medieval version of the method. For example, to convert 20/53 to an Egyptian fraction series, Leonardo selected 18/48, 3/8 raised to a multiple of 6, following a rule set down in the EMLR and the RMP, as stated as Leonardo's first method), writing the medieval answer: 20/53 = 18/48 1/8 0/53 (written in reverse order) within a notation that goes beyond the scope of this discussion. For additional details of the medieval notation, and Liber Abaci Egyptian fraction topics refer to Wikipedia and linked Egyptian fractions discussions.

Leonardo's seventh Egyptian fraction method discusses a rational number that can not be solved by one subtraction step by either following H-B or a multiple method. In this case Leonardo selected a second partition, or a second multiple, a medieval method that J.J. Sylvester in 1891 improperly reported as Fibonacci's n-step greedy algorithm.

In 2009, RMP 36 was translated to show that 30/53,28/53,15/53, 5/53, 3/53 and 2/53 were converted to optimized, but not optimal, unit fraction series, by using H-B aliquot part additive numerators. The additive numerators were written in red in RMP 36, detailing Ahmes' 2/n table method by the 2/53 conversion method.

CONCLUSION Sigler's 2002 publication of the Liber Abaci cites seven multiple methods that partitioned rational numbers into elegant Egyptian fraction series. The Liber Abaci methods connect to 3,200 older RMP 2/n table methods. Egyptian and medieval scribes considered p and q as prime numbers. Modern number theory through Hultsch-Bruins and other historians began to decode the RMP 2/n table, the Kahun 2/n table, and aspects of other Egyptian fraction texts before 1900. Yet, the 4,000 year old Egyptian texts were reported as limited to additive methods (suggested by Peet, Chace, DE Smith, Neugebauer, et al) for over 100 years. The RMP 2/n table was proven in 2005 and confirmed in 2009. The method contained an innovative 'red auxiliary' LCM method, thanks to decoding clues provided by the Hultsch-Bruins method, and other proto-number theory researchers parse RMP data considering p and q as prime numbers.