Author:
Murru Nadir,Romeo Giuliano,Santilli Giordano
Abstract
AbstractContinued fractions have been introduced in the field of p-adic numbers $$\mathbb {Q}_p$$
Q
p
by several authors. However, a standard definition is still missing since all the proposed algorithms are not able to replicate all the properties of continued fractions in $$\mathbb {R}$$
R
. In particular, an analogue of the Lagrange’s Theorem is not yet proved for any attempt of generalizing continued fractions in $$\mathbb {Q}_p$$
Q
p
. Thus, it is worth to study the definition of new algorithms for p-adic continued fractions. The main condition that a new method needs to fulfill is the convergence in $$\mathbb {Q}_p$$
Q
p
of the continued fractions. In this paper we study some convergence conditions for continued fractions in $$\mathbb {Q}_p$$
Q
p
. These results allow to define many new families of continued fractions whose convergence is guaranteed. Then we provide some new algorithms exploiting the new convergence condition and we prove that one of them terminates in a finite number of steps when the input is rational, as it happens for real continued fractions.
Funder
Università degli Studi di Trento
Publisher
Springer Science and Business Media LLC
Subject
Algebra and Number Theory
Reference19 articles.
1. Barbero, S., Cerruti, U., Murru, N.: Periodic representations for quadratic irrationals in the field of $$p$$-adic numbers. Math. Comput. 90, 2267–2280 (2021)
2. Bedocchi, E.: Nota sulle frazioni continue $$p$$-adiche. Ann. Mat. Pura Appl. 152, 197–207 (1988)
3. Bedocchi, E.: Remarks on periods of $$p$$-adic continued fractions. Bollettino dell’U.M.I. 7, 209–214 (1989)
4. Browkin, J.: Continued fractions in local fields. I. Demonstr. Math. 11, 67–82 (1978)
5. Browkin, J.: Continued fractions in local fields. II. Math. Comput. 70, 1281–1292 (2000)
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