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EnglishThe classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to p-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the p-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of p-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin p-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every n≥1, there exist infinitely many square roots of integers with periodic Browkin expansion of period 2^n, extending a previous result of Bedocchi obtained for n=1.
On periodicity of p-adic Browkin continued fractions / Capuano, Laura; Murru, Nadir; Terracini, Lea. - ELETTRONICO. - (2020).
| Titolo: | On periodicity of p-adic Browkin continued fractions |
| Autori: | |
| Data di pubblicazione: | 2020 |
| Abstract: | The classical theory of continued fractions has been widely studied for centuries for its important properties of good approximation, and more recently it has been generalized to p-adic numbers where it presents many differences with respect to the real case. In this paper we investigate periodicity for the p-adic continued fractions introduced by Browkin. We give some necessary and sufficient conditions for periodicity in general, although a full characterization of p-adic numbers having purely periodic Browkin continued fraction expansion is still missing. In the second part of the paper, we describe a general procedure to construct square roots of integers having periodic Browkin p-adic continued fraction expansion of prescribed even period length. As a consequence, we prove that, for every n≥1, there exist infinitely many square roots of integers with periodic Browkin expansion of period 2^n, extending a previous result of Bedocchi obtained for n=1. |
| Appare nelle tipologie: | 5.12 Altro |
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| File | Descrizione | Tipologia | Licenza | |
|---|---|---|---|---|
| 11.periodicity p-adic BCF.pdf | 1. Preprint / Submitted Version | PUBBLICO - Tutti i diritti riservati | Visibile a tuttiVisualizza/Apri |
http://hdl.handle.net/11583/2849213
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