Norms of indecomposable integers in real quadratic fields

TL;DR

This paper investigates the properties of indecomposable integers in real quadratic fields, providing estimates for their norms and disproving a prior conjecture about their maximal size.

Contribution

It introduces a method to estimate the norm of indecomposable integers using power series expansions and refutes a previous conjecture on their maximal norms.

Findings

Established bounds for the norm of indecomposable integers.

Disproved the conjecture of Jang-Kim regarding maximal norms.

Provided a new analytical approach using continued fraction expansions.

Abstract

We study totally positive, additively indecomposable integers in a real quadratic field $\mathbb Q(\sqrt D)$. We estimate the size of the norm of an indecomposable integer by expressing it as a power series in $u_i^{-1}$, where $\sqrt D$ has the periodic continued fraction expansion $[u_0, u_1, u_2, \dots, u_{s-1}, 2u_0, u_1, u_2, \dots]$. This enables us to disprove a conjecture of Jang-Kim [JK] concerning the maximal size of the norm of an indecomposable integer.