lean_workbook_plus_590

The positive integers $x, y$ and $z$ , where $x < y$ , satisfy $x^{3}+y^{3} = kz^{3}$ , (*) where $k$ is a given positive integer. In the case $x+y=k$ , show that $z^{3} = k^{2}-3kx +3x^{2}$ . Deduce that $(4z^{3}-k^{2})/3$ is a perfect square and that $\frac{1}{4}k^{2} \leq z^{3} <k^{2}$ . Use these results to find a solution of (*) when $k=20$ .