Fibonacci's algorithm for Egyptian fractions

I found this exercise in John Stillwell's 'Elements of Number Theory' interesting because it involves a process of descent which I have heard about but am not familiar with. The aim is to represent any fraction of the form b/a where b<a as a sum of distinct terms 1/n called unit fractions (here a, b, and n are natural numbers). Apparently the ancient Egyptians represented fractions as sums of such unit fractions in this way.

To express a fraction such as 7/15 as a sum of unit fractions we use the following method of Fibonacci. First, subtract the largest unit fraction less than 7/15 from 7/15. We have
$$\frac{7}{15}-\frac{1}{3}=\frac{2}{15}   ...(1)$$

Then subtract the largest unit fraction less than 2/15 from 2/15 so

$$\frac{2}{15}-\frac{1}{8}=\frac{1}{120}   ...(2)$$
Hence we can write
$$\frac{7}{15}=\frac{1}{3}+\frac{1}{8}+\frac{1}{120}$$
and this is in the form that we want. If we consider the numerators of the fractions that are produced by this algorithm (the RHS of equations (1) and (2)) then these numerators (2 and 1 in this case) always form a strictly decreasing sequence ending in 1. I will try and show the proof of this in another blog.

Sometimes it isn't obvious what the largest unit fraction less than a fraction is but a bit of manipulation will help. Suppose we wanted the smallest n such that

$$\frac{1}{n}<\frac{3}{71}$$

then this is equivalent to

$$n>\frac{71}{3}=23\frac{2}{3}$$
So here n=24.

Notice that I have started using Mathjax to display these equations.