Indecomposable integers in real quadratic fields

摘要

In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q (root D) where D > 1 is a squarefree integer. Their conjecture was later disproved by Kala for D 2 mod 4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D 1, 2,3 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 (root D). (C) 2019 Elsevier Inc. All rights reserved.