The Universe of Discourse

Did Ahmes find the best expansions for 2/n?

A couple of years back I was discussing the Rhind Mathematical Papyrus (RMP). It includes a table expressing !!\frac 2n!! as a sum $$\frac1{a_1}+\frac1{a_2}+\dots+\frac1{a_k} $$ fractions with numerator 1 (“unit fractions”). I said:

Getting the table of good-quality representations of !!\frac 2n!! is not trivial, and requires searching, number theory, and some trial and error. It's not at all clear that !!\frac2{105}=\frac1{90} + \frac1{126}!!.

Today I wondered: did Ahmes (the author) have the best possible expansions for all the !!\frac2n!! values, or were there some improvements the Egyptians had missed?

It turns out, yes! Or rather, maybe!

In On the Egyptian method of decomposing !!2/n!! into unit fractions the author, Abdulrahman A. Abdulaziz, points out that for !!\frac2{95}!! the Rhind Mathematical Papyrus gives the expansion $$\frac2{95} = \frac1{60} + \frac1{380} + \frac1{570}$$

but !!\frac1{380} + \frac1{570} = \frac1{288}!! so it could have been written as $$\frac2{95} = \frac1{60}+\frac1{288}.$$

But wait, maybe that wasn't an error. The Egyptians, like everyone, often had to multiply by 10. (In fact, the RMP itself, right after its !!\frac 2n!! table, has a shorter table of expansions of !!\frac n{10}!!.) And !!\frac1{60} + \frac1{380} + \frac1{570}!! is trivially multiplied by 10, whereas !!\frac1{288}!! isn't. There is some indication that Ahmes preferred fractions with even denominators, because they are easier to double, and the usual Egyptian method of multiplication required repeated doubling. But the Egyptians also sometimes decupled while multiplying, and the !!\frac1{60} + \frac1{380} + \frac1{570}!! expansion would have made both of those easy.

The methods by which Ahmes chose the expansions of !!\frac 2n!!, and the criteria by which he preferred one to another, are still unknown; he doesn't explain them. So it's tough to say that any item was or wasn't “best” from Ahmes' point of view.

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