Integers that are sums of two cubes in the cyclotomic $\mathbb{Z}_p$-extension

TL;DR

This paper investigates the representation of integers as sums of two cubes within cyclotomic $ ext{Z}_p$-extensions, showing that certain integers not expressible as sums of rational cubes also cannot be expressed as such in these extensions.

Contribution

It establishes new results on the impossibility of representing certain integers as sums of two cubes in cyclotomic $ ext{Z}_p$-extensions, extending classical results to these infinite extensions.

Findings

Certain integers not sums of rational cubes cannot be sums of two cubes in cyclotomic $ ext{Z}_p$-extensions

Results apply to large families of prime cyclic extensions of $ ext{Q}$

Provides conditions under which sums of two cubes are impossible in these extensions

Abstract

Let $n$ be a cubefree natural number and $p\geq 5$ be a prime number. Assume that $n$ is not expressible as a sum of the form $x^3+y^3$, where $x,y\in \mathbb{Q}$. In this note, we study the solutions (or lack thereof) to the equation $n=x^3+y^3$, where $x$ and $y$ belong to the cyclotomic $\mathbb{Z}_p$-extension of $\mathbb{Q}$. As an application, consider the case when $n$ is not a sum of rational cubes. Then, we prove that $n$ cannot be a sum of two cubes in certain large families of prime cyclic extensions of $\mathbb{Q}$.