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The geometry of Gaussian integer continued fractions.
- Source :
-
Journal of Number Theory . Apr2019, Vol. 197, p145-167. 23p. - Publication Year :
- 2019
-
Abstract
- Abstract The geometry of integer continued fractions and in particular, simple continued fractions has been recorded by exploring the underlying relationship | a d − b c | = 1 for a , b , c , d integers as it arises in the Farey tessellation of the hyperbolic plane H 2 and the array of Ford circles in the upper-half of the real plane R 2. Simple continued fractions may also be represented as a path on a graph whose vertices are reduced rationals and on a dual graph with vertices that are the Farey triangles in the tessellation of H 2 under the modular group. This paper produces an analogue of the above results for Gaussian integer continued fractions by examining the condition | α γ − β δ | = 1 for α , β , γ , δ Gaussian integers. Through this exploration it is possible to extend the concept of Farey neighbors to Gaussian rationals, introduce Farey sum sets, and establish the Farey tessellation of H 3 by Farey octahedrons under the action of the Picard groups without reference to the fundamental domains of the groups. A geodesic algorithm to extract a Gaussian integer continued fraction for complex numbers is introduced that is a geometrical analogue of the simple continued fraction for real numbers. [ABSTRACT FROM AUTHOR]
- Subjects :
- *CONTINUED fractions
*INTEGERS
*MATHEMATICS
*RATIONAL numbers
*COMPOSITE numbers
Subjects
Details
- Language :
- English
- ISSN :
- 0022314X
- Volume :
- 197
- Database :
- Academic Search Index
- Journal :
- Journal of Number Theory
- Publication Type :
- Academic Journal
- Accession number :
- 133557256
- Full Text :
- https://doi.org/10.1016/j.jnt.2018.08.006