The ancient Egyptians expressed all fractions as
sums of fractions with unit numerators. For example, they might express
as .
(In point of fact, the Egyptians also had a symbol for the fraction
but to keep things simple this will be ignored.) It should be noted that
not all fractions have a unique Egyptian representations. One way of obtaining
an Egyptian representation of a fraction is known as the *Greedy Algorithm*.
This algorithm simply adds to the sum so far the largest possible unit
fraction which does not make the sume exceed the given fraction. For example,
to find the Egyptian represention of
note that
but
so start with .
Then consider .
Note that
but that .
Hence the second term provided by the Greedy Algorith is .
Next, consider .
The Greedy Algorithm therefore yields .

Implement the Greedy Algorithm in a procedure which calculates the Egyptian
representation of any function. In order to do this, consider the problem
of finding the largest fraction with unit numerator less than or equal
to .
Find an efficient way of doing this with the Maple function `ceil`.
A good strategy for your procedure would be to use recursion. One way of
doing this is to define a procedure, which we will call Egyptian, which
takes two inputs. The first input is a list of the fractions obtained so
far by the Greedy Algorithm and the second variable is the fraction to
be represented. So, for example, to find the representation of
you would type `Egyptian([],3/7)` and the output would be the list .

What Egyptian representation does your procedure yield for the fraction
Does the Greedy Algorithm always yield the shortest possible Egyptian represenation
of a fraction? Does it always yield the smallest denominators? One way
of getting other representations for a fraction such as
is to first split it as a sum -- for example,.
Write a procedure wich finds the Egyptian decompositions for
using the Greedy Algorithm for both sides of the sum
where *A*+ *B* = 65 and both *A* and *B* are positive.

A question which has not been addressed is why the Greedy Algorithm is guaranteed to stop. Might there not be a fraction which results in the Greedy Algorithm computing an infinite sequence of increasingly smaller fractions with unit numerator? The negative answer to this question was established by Fibonacci in 1202. How did he prove this? To provide at least a partial answer to this question, write a procedure which calculates the remainders of each step of the Greedy Algorithm and apply it to various fractions. You should be able to arrive at a conjecture about the behaviour of the sequence of numerators of these remainders. If this conjecture were true, how could it be used to establish Finonacci's theorem on the the Greedy Algorithm? Can you provide a proof of your conjecture and thereby prove Finonacci's theorem?