Abstract
In 2016, Jang and Kim stated a conjecture about the norms of indecomposable integers in real quadratic number fields Q(D) where D>1 is a squarefree integer. Their conjecture was later disproved by Kala for D≡2mod4. We investigate such indecomposable integers in greater detail. In particular, we find the minimal D in each congruence class D≡1,2,3mod4 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(D).
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