Computing with Egyptian Fractions

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?

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