论文标题
关于$ p $ adic browkin的周期性持续分数
On periodicity of $p$-adic Browkin continued fractions
论文作者
论文摘要
几个世纪以来，持续分数的经典理论因其良好近似的重要特性而被广泛研究，最近，它已被推广到$ p $ adadig的数字，在这些数字上呈现了与真实情况有关的许多差异。在本文中，我们调查了Browkin引入的$ P $ ADIC持续分数的周期性。我们提供了一些一般必要的周期性条件，尽管仍然缺少具有纯粹是周期性眉毛持续扩展的$ p $ adadic数字的完整表征。在本文的第二部分中，我们描述了一个一般程序，以构建具有定期browkin $ p $ p $ p $ p $ p $ p $ p $ p $ p $ p $ partion的整数的正方形根部的一般程序。结果，我们证明，对于每$ n \ ge 1 $，就会存在无限的许多$ \ sqrt {m} \ in \ qq_p $，并具有定期的browkin扩展$ 2^n $，却占据了Bedocchi的先前结果，以$ n = 1 $获得。
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 $n \ge 1$, there exist infinitely many $\sqrt{m}\in \QQ_p$ with periodic Browkin expansion of period $2^n$, extending a previous result of Bedocchi obtained for $n=1$.
