The paper "Norms of indecomposable integers in real quadratic fields" is on arxiv and was published in J. Number Theory 166 (2016), 193-207.
Mathematica calculations
Proof of Theorem 4: the notebook is available below as indecomposable.nb and .txt.
Argument for the existence of infinitely many counterexamples (on penultimate page): the notebook is available below as indecomposable 2.nb and .txt.