On a family of unit equations over simplest cubic fields

TL;DR

This paper completely solves a family of unit equations over simplest cubic fields for certain integer bounds, advancing understanding of units in these algebraic number fields.

Contribution

It provides a complete solution to specific unit equations in simplest cubic fields under a new bounded restriction on n.

Findings

Explicit solutions for unit equations when |n| ≤ max{1, |a|^{1/3}}

Characterization of units in simplest cubic fields

Extension of known results to a broader class of equations

Abstract

Let $a\in \mathbb{Z}$ and $\rho$ be a root of $f_a(x)=x^3-ax^2-(a+3)x-1$, then the number field $K_a=\mathbb{Q}(\rho)$ is called a simplest cubic field. In this paper we consider the family of unit equations $u_1+u_2=n$ where $u_1,u_2\in \mathbb{Z}[\rho]^*$ and $n\in \mathbb{Z}$. We completely solve the unit equations under the restriction $|n|\leq \max\{1,|a|^{1/3}\}$.