Show that if
\[\frac{1}{x} + \frac{2}{y} = \frac{2}{7}\,,\]
then \((2x - 7)(y - 7) = 49\).
By considering the factors of \(49\), find all the pairs of positive integers \(x\) and \(y\) such that
\[\frac{1}{x} + \frac{2}{y} = \frac{2}{7}\,.\]
Let \(p\) and \(q\) be prime numbers such that
\[p^2 + pq + q^2 = n^2\]
where \(n\) is a positive integer. Show that
\[(p + q + n)(p + q - n) = pq\]
and hence explain why \(p + q = n + 1\).
Hence find the possible values of \(p\) and \(q\).
Let \(p\) and \(q\) be positive and
\[p^3 + q^3 + 3pq^2 = n^3\,.\]
Show that \(p + q - n < p\) and \(p + q - n < q\).
Show that there are no prime numbers \(p\) and \(q\) such that \(p^3 + q^3 + 3pq^2\) is the cube of an integer.