The Universe of Discourse : Did Ahmes find the best expansions for 2/n?
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Tue, 17 Mar 2026
Did Ahmes find the best expansions for 2/n?
A couple of years back I was discussing the Rhind Mathematical Papyrus<br>(RMP). It includes a table expressing !!\frac 2n!! as a sum<br>$$\frac1{a_1}+\frac1{a_2}+\dots+\frac1{a_k} $$ fractions with<br>numerator 1 (“unit fractions”). I said:
Getting the table of good-quality representations of !!\frac 2n!! is not<br>trivial, and requires searching, number theory, and some trial and<br>error. It's not at all clear that !!\frac2{105}=\frac1{90} +<br>\frac1{126}!!.
Today I wondered: did Ahmes (the author) have the best possible<br>expansions for all the !!\frac2n!! values, or were there some<br>improvements the Egyptians had missed?
It turns out, yes! Or rather, maybe!
In<br>On the Egyptian method of decomposing !!2/n!! into unit fractions<br>the author, Abdulrahman A. Abdulaziz, points out that for<br>!!\frac2{95}!! the Rhind Mathematical Papyrus gives the expansion<br>$$\frac2{95} = \frac1{60} + \frac1{380} + \frac1{570}$$
but !!\frac1{380} + \frac1{570} = \frac1{228}!! so it could have been<br>written as $$\frac2{95} = \frac1{60}+\frac1{228}.$$
But wait, maybe that wasn't an error. The Egyptians, like everyone,<br>often had to multiply by 10. (In fact, the RMP itself, right after<br>its !!\frac 2n!! table, has a shorter table of expansions of !!\frac<br>n{10}!!.) And !!\frac1{60} + \frac1{380} + \frac1{570}!! is trivially<br>multiplied by 10, whereas !!\frac1{228}!! isn't. There is some<br>indication that Ahmes preferred fractions with even denominators,<br>because they are easier to double, and the usual Egyptian method of<br>multiplication required repeated doubling. But the Egyptians also<br>sometimes decupled while multiplying, and the !!\frac1{60} +<br>\frac1{380} + \frac1{570}!! expansion would have made both of those<br>easy.
The methods by which Ahmes chose the expansions of !!\frac 2n!!, and<br>the criteria by which he preferred one to another, are still unknown;<br>he doesn't explain them. So it's tough to say that any item was or<br>wasn't “best” from Ahmes' point of view.
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