# Indecomposable integers in real quadratic fields

## TL;DR

This paper investigates indecomposable integers in real quadratic fields, disproves a conjecture by Jang and Kim for certain cases, and proposes a refined conjecture with partial proof of its accuracy.

## Contribution

It identifies minimal counterexamples to the Jang-Kim Conjecture across different congruence classes and introduces a refined conjecture with proven bounds.

## Key findings

- Disproved Jang-Kim Conjecture for D ≡ 2 mod 4
- Found minimal counterexamples in each congruence class D ≡ 1,2,3 mod 4
- Proved a weaker version of the refined conjecture with bounds of order √D

## 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 $D \equiv 2 \bmod 4$. We investigate such indecomposable integers in greater detail. In particular, we find the minimal $D$ in each congruence class $D \equiv 1,2,3 \bmod 4$ 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 $O \left( \sqrt{D} \right)$.