2020/10/14 by Capuano, Laura, Murru, Nadir, Terracini, Lea · 1 citation
#11D88 #11J70 #11Y65 #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.2010.07364
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 √(m)∈ \QQp with periodic Browkin expansion of period 2n, extending a previous result of Bedocchi obtained for n=1.