farey -子图与连分数

Pub Date : 2021-06-28 DOI:10.1556/012.2022.01525

S. Kushwaha, R. Sarma

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引用次数: 3

摘要

在这篇文章中，我们研究了Farey图的一组子图，对于每一个N∈N，我们将其记为_ (N)。我们证明了当且仅当N等于1或一个素数幂时，N是连通的。我们引入一类连分数，称为_ (N) -对于每个N > 1的连分数。建立了图_ _ N中_ _ N连分式与无穷远处若干路径之间的关系。利用这种对应关系，讨论了实数的连分数展开式的存在唯一性。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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Farey-Subgraphs and Continued Fractions

In this article, we study a family of subgraphs of the Farey graph, denoted as ℱN for every N ∈ ℕ. We show that ℱN is connected if and only if N is either equal to one or a prime power. We introduce a class of continued fractions referred to as ℱN -continued fractions for each N > 1. We establish a relation between ℱN-continued fractions and certain paths from infinity in the graph ℱN. Using this correspondence, we discuss the existence and uniqueness of ℱN-continued fraction expansions of real numbers.

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