A number of the form \(1/N\), where \(N\) is an integer greater than \(1\), is called a unit fraction.
Noting that \[\frac{1}{2} = \frac{1}{3} + \frac{1}{6}\quad\mbox{and}\quad\frac{1}{3}=\frac{1}{4}+\frac{1}{12},\] guess a general result of the form \[\frac{1}{N} = \frac{1}{a}+\frac{1}{b}\] and hence prove that any unit fraction can be expressed as the sum of two distinct unit fractions.
By writing the previous equation in the form \[(a − N)(b − N) = N^2\] and by considering the factors of \(N^2\), show that if \(N\) is prime, then there is only one way of expressing \(1/N\) as the sum of two distinct unit fractions.
Prove similarly that any fraction of the form \(2/N\), where \(N\) is prime number greater than \(2\), can be expressed uniquely as the sum of two distinct unit fractions.
