On integers that are representable as the sum of two units

Abstract

Let $K$ be a number field of degree $D$ with maximal order ${\mathcal{O}}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D\ge 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in ${\mathcal{O}}_K^{\ast }$ is a finite effectively computable set. This result partially resolves an open problem posed by Tinková, Yatsyna, and the first author. We illustrate our method by explicitly computing $N_K$ for the smallest totally real quintic field $K$ with Galois group $S_5$.

20 pages; comments are welcome!