  In this note I give simple proofs of classical results of Euler, Legendre and
Sylvester showing that for certain integers M there are no (or only a few)
solutions of $x^3 + y^3 = M$, with $x$ and $y$ in $\mathbb{Q}$. The proofs all
use a single argument -- infinite 3-descent in the ring $\mathcal{O} =
\mathbb{Z}[\omega]$ of Eisenstein integers. (Everything needed about
$\mathcal{O}$ is developed from scratch.) The reader only needs the briefest
acquaintance with complex numbers, fields and congruence modulo an element of a
commutative ring. In particular I never say anything about ideals or elliptic
curves (though I do mention cubic reciprocity in passing), and a clever
high-school student might well enjoy the note. A few new results with $M$ in
$\mathcal{O}$ and $x$ and $y$ in $\mathbb{Q}[\omega]$ are also derived.
