Sums of two units in number fields

TL;DR

This paper investigates the set of positive integers that can be expressed as the sum of two units in the ring of integers of a number field, providing finiteness results and explicit classifications in specific cases.

Contribution

It proves finiteness of such sums in certain number fields and classifies solutions for cubic fields with specific properties.

Findings

The set of such integers is finite if the field lacks a real quadratic subfield.

Explicit solutions are classified for cyclic cubic fields.

Solutions are characterized for cubic fields with negative discriminant.

Abstract

Let $K$ be a number field with ring of integers $\mathcal{O}_K$. Let $\mathcal{N}_K$ be the set of positive integers $n$ such that there exist units $\varepsilon, \delta \in \mathcal{O}_K^\times$ satisfying $\varepsilon + \delta = n$. We show that $\mathcal{N}_K$ is a finite set if $K$ does not contain any real quadratic subfield. In the case where $K$ is a cubic field, we also explicitly classify all solutions to the unit equation $\varepsilon + \delta = n$ when $K$ is either cyclic or has negative discriminant.