INTRODUCTION: The Rhind Mathematical Papyrus, written by Ahmes in 1650 BCE, and the Kahun Papyrus, written 200 years earlier, began with 2/n tables. Scholars have debated the topic for 130 years. How did scribes convert 2/p and 2/n to optimal or elegant Egyptian fraction series?

The first number theory solution to the 2/p problem was suggested by F. Hultsch in 1895. Hultsch considered even denominator first partitions available between p/2 and p. Each denominator is 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 are considered. Ahmes selected 1/12. The 1/12 denominator contains aliquot parts: 12, 6, 4, 3, 2 and 1, such that 2/19 - 1/12 creates a remainder 5/(12*19). The remainder's numerator 5 = (3 + 2) was additively obtained by ignoring the smaller last term alternative(s), in this case 5= (4 + 1). Ahmes consistently, but not always, reported optimal or elegant Egyptian fraction series.

Ahmes' optimal method was likely reduced to a mental process by considering aliquot parts of the divisor of the first partition by considering a multiple. Today, composite and primes factor first partition denominators are optimally solved by writing n/p and n/pq, picking an optimal first partition and creating a remainder by a subtraction step. Almost anyone can mentally work this class of problem with a little practice. Please note that the Hultsch-Bruins method is a code breaking technique. Not all of Ahmes' 2/n conversions can be reproduced applying the simple H-B rules. A recently reported medieval multiple method does calculate all 51 RMP 2/n table Egyptian fraction series.

Before discussing a medieval multiple method that was likely used by Ahmes let's look at 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 auxilinary' numbers were applied such that 2/19 = 1/12 + 5/(12*19).

Ahmes may have asked, can the numerator 5 be found by selecting two or more divisors of 12? In descending order, Ahmes considered two solutions (4 + 1) and (3 + 2). Ahmes selected (3 + 2) as noted by: 2/19 = 1/12 + (3+2)/(12*19) = 1/12 + 1/76 1/114. Note that optimal first partitions and optimal multiples solve 2/p conversion problems. Ahmes used red auxiliary numbers by sorting alternative solutions between p/2 and p.

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 + (35 + 14)/(70*91) = 1/70 + 1/182 + 1/455

As a medieval alternative, that could have been passed down through 2,800 years, though Egyptian, Greek, Hellene, and Arab cultures, Ahmes would have selected a single multiple method that converted all 2/n table members. Yes, a single method. Since the multiple method also describes the 26 series reported in the Egyptian Mathematical Leather Roll, there is a high likely-hood that Ahmes used the method.

In 1944, E. M. Bruins independently (no knowledge of Hultsch's work) verified Hultsch's 49 year old solution to the 2/th table problem. Even though math historians have debated the historical validity of the Hultsch-Bruin method for over 60 years, no acceptance or rejection of the method has been formally reached.

The classical view that Egyptian arithmetic was only additive has held back professional journals from formally adopting Hultsch-Bruins. The same fate may effect the professional status of the newly reported optimal multiple method for a few years. In the long run, the following will be considered.

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 discussions changed the RMP 2/n table debate. The Liber Abaci showed that the Hultsch-Bruins method was not generally used by Ahmes to find all 2/n table Egyptian fraction series.

The Liber Abaci summarizes three versions of the Hultsch-Bruins method. The first two definitions create ancient Egyptian fraction series, meaning that Ahmes likely used. 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. Yet, a multiple method dominates 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, as stated as Leonardo's first method), writing the medieval answer: 20/53 = 18/48 1/8 0/53 (actually 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 new medieval method that J.J. Sylvester in 1891 improperly reported as related to the greedy algorithm.

CONCLUSION The 2002 publication of the Liber Abaci reveals seven multiple methods that exactly partition 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 had considered p and q as prime numbers. Modern number theory through Hultsch-Bruins began to decode the RMP 2/n table, the Kahun 2/n tables and aspects of other Egyptian fraction texts as written by ancient scribes. The 4,000 year old Egyptian texts were therefore not limited to additive methods as suggested by Peet, Chace, DE Smith, Neugebauer, et al, in the 1920's. The RMP 2/n table problem has therefore been solved by finding Ahmes' optimal multiple method in medieval texts, thanks to clues provided by the Hultsch-Bruins method and proto-number theory methods that considered p and q as prime numbers.