
Periodic representations for quadratic irrationals in the field of đť‘ť-adic numbers
Author(s) -
Stefano Barbero,
Umberto Cerruti,
Nadir Murru
Publication year - 2021
Publication title -
mathematics of computation
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.95
H-Index - 103
eISSN - 1088-6842
pISSN - 0025-5718
DOI - 10.1090/mcom/3640
Subject(s) - algorithm , annotation , type (biology) , mathematics , computer science , artificial intelligence , biology , ecology
Continued fractions have been widely studied in the field of p p -adic numbers Q p \mathbb Q_p , but currently there is no algorithm replicating all the good properties that continued fractions have over the real numbers regarding, in particular, finiteness and periodicity. In this paper, first we propose a periodic representation, which we will call standard , for any quadratic irrational via p p -adic continued fractions, even if it is not obtained by a specific algorithm. This periodic representation provides simultaneous rational approximations for a quadratic irrational both in R \mathbb R and Q p \mathbb Q_p . Moreover given two primes p 1 p_1 and p 2 p_2 , using the Binomial transform, we are also able to pass from approximations in Q p 1 \mathbb {Q}_{p_1} to approximations in Q p 2 \mathbb {Q}_{p_2} for a given quadratic irrational. Then, we focus on a specific p p –adic continued fraction algorithm proving that it stops in a finite number of steps when processes rational numbers, solving a problem left open in a paper by Browkin [Math. Comp. 70 (2001), pp. 1281–1292]. Finally, we study the periodicity of this algorithm showing when it produces standard representations for quadratic irrationals.