On a family of unit equations over simplest cubic fields

Abstract

Let $a\in \mathbb{Z}$ and $\rho$ be a root of $f_a(x)=x^3-a{x}^2-(a+3)x-1$, 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 $u_1,u_2\in \mathbb{Z}[\rho {]}^{\ast }$ and $n\in \mathbb{Z}$. We completely solve the unit equations under the restriction $|n|\le \max \{1,|a{|}^{1/3}\}$.