Abstract

Let $M$ be a cubefree integer divisible by precisely two primes, just one of which, $p$, is $1\mathrm{\text{ mod }}3$. Theorem XI of Selmer (1951) shows that certain such $M$ cannot be written as $x^3+y^3$ with $x$ and $y$ in $\mathbb{Q}$. The theorem has two parts, labeled as (9.8.5) and (9.8.6).

In (9.8.5), $M\equiv 3$ or $6 (9)$ so that the second prime dividing $M$ is $3$, and $M$ is $3p$ or $3p^2$; the result holds when $3$ is not a cube in $\mathbb{Z}∕p$.

In (9.8.6), $M\equiv 2,4,5$ or $7 (9)$ so that the second prime dividing $M$ is some $q\equiv -1 (3)$ and $M$ is $pq,p{q}^2,p^{2}q$ or $p^2{q}^2$; the result holds when $q$ is not a cube in $\mathbb{Z}∕p$.

In this note we give short proofs of (9.8.5) and (9.8.6), using only easy facts about the ring $\mathcal{𝒪}$ of Eisenstein integers together with one assertion that follows from cubic reciprocity. Write $p$ as $\pi \overline{\pi }$ in $\mathcal{𝒪}$ with $\pi \equiv 1 (3)$; then $\overline{\pi }$ is a cube in $\mathcal{𝒪}∕\pi$. This was shown by Gauss, by Eisenstein, and by Jacobi. But once $M$ (and thus $p$) is chosen, the proof of the assertion is just a brief calculation.

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