Indecomposable integers in real quadratic fields

Abstract

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields $\mathbb{Q}\left(\sqrt{D}\right)$ where $D>1$ is a squarefree integer. Their conjecture was later disproved by Kala for $D\equiv 2\mathrm{mod}4$. We investigate such indecomposable integers in greater detail. In particular, we find the minimal $D$ in each congruence class $D\equiv 1,2,3\mathrm{mod}4$ that provides a counterexample to the Jang-Kim Conjecture; provide infinite families of such counterexamples; and state a refined version of the Jang-Kim conjecture. Lastly, we prove a slightly weaker version of our refined conjecture that is of the correct order of magnitude, showing the Jang-Kim Conjecture is only wrong by at most $O\left(\sqrt{D}\right)$.

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• Arithmetic of cubic number fields: Jacobi-Perron, Pythagoras, and indecomposables
• Additive structure of non-monogenic simplest cubic fields