Red auxiliary numbers, from Wikipedia

In the study of ancient Egyptian math include red additive numerator series. The majority of Rhind Mathematical Papyrus problems report these red numbers. Only in RMP 36 were the intermediate red auxiliary numerators specifically listed. The summed red numbers are numerators of a scaled rational number (i.e. 53 + 4 + 2 + 1)/1060 = 60/1060 = 3/53*(20/20), one of the conversions listed in RMP 36.

Ahmes also practiced selections of LCMs (m/m), i.e. 20/20 to convert 3/53, in three RMP problems in RMP 21-23.

Red auxiliary numbers were noted by historians for 130 years. One purpose allowed 2/n table rational numbers to be converted to optimized, but not always optimal, Egyptian fraction series. Math historians failed to parse most basic of the red auxiliary number applications until the 21st century. Early math historians only recognized that these numbers were connected to LCMs.

It took over 120 years years to parse the red auxiliary numbers in the RMP's 84 problems. As a review, Ahmes' three practice red auxiliary problems, George G. Joseph, "Crest of the Peacock", cites fragmented details from which ancient scribal students learned to apply the method. On page 37, example 3.7 Joseph reports:

Complete 2/3 + 1/4 + 1/28 to 1.

This meant: solve $$2/3 + 1/4 + 1/28 + x = 1$$ (example 3.7)

The lowest common denominator (LCM) was not 28, or 84, but rather 42. Unskilled modern students would multiple 3 times 28 finding an LCM of 84. But 42 was sufficient for Egyptian scribes as noted by:

$$84/3 + 42/4 + 42/28 + 42x = 42$$ (example 3.7.1)

as written in fractions

$$28 + (10 + 1/2) + (1 + 1/2) + 2 = 42$$ (example 3.7.2)

with 42 marked in red. Modern algebra would have included 42x, solving

$$42x = 2$$

$$x = 2/42 = 1/21$$ (example 3.7.3)

meaning scribes would have a written final series in the form:

$$2/3 + 1/4 + 1/28 + 1/21 = 1$$ (example 3.7.4)

The RMP included three problems that asked Ahmes to complete a series of fractions adding to a given number. Two problems, RMP 21 and RMP 23, follow:

RMP 21: Complete $$2/3 + 1/15 + x = 1$$

use red auxiliary number 30 to find

$$60/3 + 30/15 + 30x = 30$$

$$20 + 2 + 8 = 30$$

$$30x = 8$$

$$x = 8/30 = 4/15 = (3 + 1)/15 = 1/5 + 1/15$$

such that:

$$2/3 + 1/5 + 2/15 = 1$$

was written as:

$$2/3 + 1/5 + 1/10 + 1/30 = 1$$

RMP 23: Complete $$1/4 + 1/8 + 1/10 + 1/35 + 1/45 + x = 2/3$$

use red auxiliary 45 to compute $$x = 1/9 + 1/40$$

In addition, RMP 36 solved 2/53, 3/53, 5/53, 28/53 and 30/53 by finding red auxiliary numerators considering:

2/53*(30/30) = 60/1590 = (53 + 5 + 2)/1590 = 1/20 + 1/318 + 1/795

3/53*(20/20) = 60/1060 = (53 + 4 + 2 + 1)/1060 = 1/20 + 1/265 + 1/530 + 1/1060

5/53*(12/12) = 60/636 = (53 + 4 + 2 + 1)/636 = 1/12 + 1/159 + 1/318 + 1/636

28/53*(2/2) = 56/106 = (53 + 2 + 1)/106 = 1/2 + 1/53 + 1/106

and,

30/53 = 2/53 + 28/53

with numerators (53 + 5 + 2), (53 + 4 + 2 + 1) and (53 + 2 + 1) written in red.

Note that 30/53 can not be solved by finding one LCM integer. In RMP 31, 28/97 also can not be solved by finding one LCM. In both cases the 2/n table allowed conversions of 30/53 and 28/97 to optimized unit fraction series to be found. In the 30/53 case, 30/53 became 2/53 + 28/53. In the 28/96 case, 28/97 was parsed to 2/97 + 26/97. That is, the 2/n table allowed almost any rational number, n/p and n/pq, to be converted to an optimized, but not always optimal unit fraction series, by first considerring n/p = 2/p + (n-2)/p, and n/pq = 2/pq + (n- 2)/pq.

It turns out that the Egyptian Mathematical Leather Roll (EMLR) and the RMP 2/n table employed LCM's to convert rational numbers to Egyptian fraction series. Interestingly, the EMLR used non-optimal LCMs allowing students to select any LCM guess and work out an Egyptian fraction series. More importantly, the RMP began with 1/3 of the text reporting 51 2/n optimized, but not always optimal, Egyptian fraction series.

That is, red auxiliary numbers defined a core method that math historians pondered for 130 years, but did not see. Visibility began to appear around 2002 with publications of the EMLR, the Akhmim Wooden Tablet, the Ebers Papyrus, and other Egyptian fraction texts.

Understanding Ahmes' 'red auxiliary' numbers, and its proto-number theory properties, allows RMP arithmetic methods to come into focus. Recent journal papers report Egyptian fraction arithmetic in updated ways. One of the more important updates reports scribal multiplication and division of rational numbers as inverse operations. The ancient arithmetic operations had not followed duplation multiplication, as proposed for over 100 years, but the ancient operations in RMP 38, and other problems, looked and acted more like modern multiplication and division operations.

Three 2001 Russian math encyclopedia entries discuss this Egyptian fractions topic beginning with a modern definition of LCMs. The encyclopedia was published on-line by Springer, with one entry suggesting ancient LCMs were known to Ahmes by writing:

$$3/11 = 1/6 + 1/11 + 1/66$$

and, converting

$$3/11* (6/6) = (18/66) = (11 + 6+ 1)/66 = 1/6 + 1/11+ 1/66$$

The first entry mentions modern LCMs, and did not attempt to take an ancient leap back in time. Had a formal hypothesis been offered a clear path from the present to a 4,000 year old ancient academic discussion would have be set in an interdisciplinary context. Proof, or disproof, that Ahmes thought in a modern version LCMs could then be formally presented.

A third Russian entry generally reports an ancient aliquot fraction or ratio idea (Russian terms for red auxiliary numbers) without identifying a hypothetical use of ancient LCMs.

Considering the three Russian entries, Ahmes' three LCM practice problems, and Ahmes' 2/n table patterns as one topic, a clear outline of a proof that Ahmes did use a modern LCM definition is provided.