Finiteness and periodicity of continued fractions over quadratic number fields
The result's identifiers
Result code in IS VaVaI
<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F68407700%3A21340%2F22%3A00355955" target="_blank" >RIV/68407700:21340/22:00355955 - isvavai.cz</a>
Result on the web
<a href="https://doi.org/10.24033/bsmf.2845" target="_blank" >https://doi.org/10.24033/bsmf.2845</a>
DOI - Digital Object Identifier
<a href="http://dx.doi.org/10.24033/bsmf.2845" target="_blank" >10.24033/bsmf.2845</a>
Alternative languages
Result language
angličtina
Original language name
Finiteness and periodicity of continued fractions over quadratic number fields
Original language description
We consider continued fractions with partial quotients in the ring of integers of a quadratic number field K and we prove a generalization to such continued fractions of the classical theorem of Lagrange. A particular example of these continued fractions is the β-continued fraction introduced by Bernat. As a corollary of our theorem we show that for any quadratic Perron number β, the β-continued fraction expansion of elements in Q(β) is either finite of eventually periodic. The same holds for β being a square root of an integer. We also show that for certain 4 quadratic Perron numbers β, the β-continued fraction represents finitely all elements of the quadratic field Q(β), thus answering questions of Rosen and Bernat. Based on the validity of a conjecture of Mercat, these are all quadratic Perron numbers with this feature.
Czech name
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Czech description
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Classification
Type
J<sub>imp</sub> - Article in a specialist periodical, which is included in the Web of Science database
CEP classification
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OECD FORD branch
10101 - Pure mathematics
Result continuities
Project
<a href="/en/project/EF16_019%2F0000778" target="_blank" >EF16_019/0000778: Center for advanced applied science</a><br>
Continuities
P - Projekt vyzkumu a vyvoje financovany z verejnych zdroju (s odkazem do CEP)
Others
Publication year
2022
Confidentiality
S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů
Data specific for result type
Name of the periodical
Bulletin de la Société Mathématique de France
ISSN
0037-9484
e-ISSN
2102-622X
Volume of the periodical
150
Issue of the periodical within the volume
1
Country of publishing house
FR - FRANCE
Number of pages
33
Pages from-to
77-109
UT code for WoS article
000853257900003
EID of the result in the Scopus database
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