A number of the form , where is an integer greater than 1, is called a unit fraction.
guess a general result of the form
and hence prove that any unit fraction can be expressed as the sum of two distinct unit fractions.
By writing in the form
and by considering the factors of , show that if is prime, then there is only one way of expressing as the sum of two distinct unit fractions.
Prove similarly that any fraction of the form , where is prime number greater than 2, can be expressed uniquely as the sum of two distinct unit fractions.
Fractions written as the sum of unit fractions are called Egyptian fractions: they were used by Egyptians. The earliest record of such use is 1900BC. The Rhind papyrus in the British Museum gives a table of representations of fractions of the form as sums of unit fractions for all odd integers between 5 and 101 — a remarkable achievement when you consider that algebra was 3,500 years in the future.
It is not clear why Egyptians represented fractions this way; maybe it just seemed a good idea at the time. Certainly the notation they used, in which for example was denoted by with an oval on top, does not lend itself to generalisation to fractions that are not unit. One of the rules for expressing non-unit fractions in terms of unit fractions was that all the unit fractions in should be distinct, so had to be expressed as instead of as , which seems pretty daft.
It is not obvious that every fraction can be express in Egyptian form; this was proved by Fibonacci in 1202. There are however still many unsolved problems relating to Egyptian fractions.
Egyptian fractions have been called a wrong turn in the history of mathematics; if so, it was a wrong turn that favoured style over utility; no bad thing, in my opinion.
Solution to problem 11
A good guess would be that the ﬁrst term of the decomposition of is , i.e. . In that case, the other term is i.e. . That proves the result that every unit fraction can be expressed as the sum of two unit fractions.
The only factors of (since is prime) are , and . The possible factorisations of are therefore or . We discount , since this would lead to , and this is ruled out in the question. That leaves only and (or the other way round). Thus and are the only possible values for and and the decomposition is unique.
For the second half, set
where . We need to aim for an equation with on one side, so that we can use the method of the ﬁrst part. We have which we write as
Thus and (or the other way round). The decomposition is therefore unique and given by
The only possible ﬂy in the ointment is the in the denominators: and are supposed to be integers. However, all prime numbers greater than 2 are odd, so and are both even and the denominators are indeed integers.
Another way of getting the last part (less systematically) would have been to notice that
which gives the result immediately. How might you have noticed this? Well, I noticed it by trying to work out some examples, starting with what is given at the very beginning of the question. If then which works.
Of course, the advantage of the systematic approach is that it allows generalisation: what happens if is odd but not prime; what happens if the numerator is 3 not 2?