Distribution of periods of the continued fractions for quadratic irrationals

Tlepova M.

Distribution of periods of the continued fractions for quadratic irrationals

dc.contributor.author	Tlepova M.
dc.date.accessioned	2025-06-13T05:01:40Z
dc.date.available	2025-06-13T05:01:40Z
dc.date.issued	2023
dc.description.abstract	This dissertation thesis focuses on the distribution patterns of the periods of continued fractions for quadratic irrational numbers. The main objective of this thesis is to investigate and examine the patterns and distribution of the periods of continued fractions associated with quadratic irrational numbers of the form a = nVd, where n is a positive integer and d is a square-free integer. We denote the length of the period of the continued fraction as D(a). We establish a relationship between the length of the periodic part of the continued fraction D(a) and the quadratic irrational number a = nvd by introducing an exact and explicit formula for the irrational number a for specific cases when D(a) = 2 and D(a) = 3. By utilizing the equation a = n d and considering cases where the length of the periodic part is 2, we have successfully demonstrated that when n 2 1 is the solution of the Pell’s equation in the form of n? — dz* = 1, the corresponding quadratic irrational a = nvd will indeed have a periodic part of length 2. Through the proof of this theorem, we have established that for any valuc of d greatcr than or equal to 1, there exists an infinite set of positive integers n for which the length of the period D(nv 4) is precisely 2.
dc.identifier.citation	Tlepova M / Distribution of periods of the continued fractions for quadratic irrationals / 7M05401 - Department of Mathematics / 2023
dc.identifier.uri	https://repository.sdu.edu.kz/handle/123456789/1759
dc.language.iso	en
dc.publisher	Faculty of Engineering and Natural Sciences
dc.title	Distribution of periods of the continued fractions for quadratic irrationals
dc.type	Thesis