Indecomposable integers in real quadratic fields
| Autor: | Magdaléna Tinková, Paul Voutier |
|---|---|
| Rok vydání: | 2020 |
| Předmět: |
Algebra and Number Theory
Conjecture Mathematics - Number Theory Mathematics::Number Theory 010102 general mathematics 010103 numerical & computational mathematics State (functional analysis) Square-free integer Algebraic number field 01 natural sciences Physics::History of Physics Combinatorics Quadratic equation Integer FOS: Mathematics Number Theory (math.NT) 0101 mathematics Indecomposable module Mathematics Counterexample |
| Zdroj: | Journal of Number Theory. 212:458-482 |
| ISSN: | 0022-314X |
| DOI: | 10.1016/j.jnt.2019.11.005 |
| Popis: | 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 ≡ 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 ( D ) . |
| Databáze: | OpenAIRE |
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