Norms of indecomposable integers in real quadratic fields

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.