Norms of indecomposable integers in real quadratic fields

Vítězslav Kala

Published 2015-12-15Version 1

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 $[u-0, \bar{u_1, u_2, \dots, u_{s-1}, 2u-0}]$. This enables us to find a counterexample to a conjecture of Jang-Kim [JK] concerning the maximal size of the norm of an indecomposable integer.