Norms of indecomposable integers in real quadratic fields

We study totally positive, additively indecomposable integers in a real quadratic field Q(D). We estimate the size of the norm of an indecomposable integer by expressing it as a power series in ui−1, where D has the periodic continued fraction expansion [u0,u1,u2,…,us−1,2u0‾]. This enables us to disprove a conjecture of Jang–Kim [JK] concerning the maximal size of the norm of an indecomposable integer.