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

TL;DR

This paper presents simple, classroom-friendly proofs of classical results on the sum of two cubes problem using infinite 3-descent in Eisenstein integers, making the topic accessible to high school students.

Contribution

It introduces an accessible approach to classical number theory results on sums of cubes, avoiding advanced concepts like elliptic curves and ideals.

Findings

Classical results on sums of two cubes are proved using elementary methods.

New results involving Eisenstein integers and rational numbers are derived.

The approach is suitable for educational purposes and high school students.

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.