The sum of two cubes problem -- an approach that's classroom friendly

Abstract

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.