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Articles 1 - 6 of 6
Full-Text Articles in Number Theory
Complex Continued Fractions And Inadmissible Sequences, Carolina A. Rizzi, Holly Vanlooy
Complex Continued Fractions And Inadmissible Sequences, Carolina A. Rizzi, Holly Vanlooy
Rose-Hulman Undergraduate Mathematics Journal
Serret's Theorem says that real numbers are related by a type of Möbius transformation if and only if the tails of their regular continued fraction expansions are the same. Serret’s Theorem does not hold for Hurwitz Continued Fractions in the complex plane due to a counterexample of Lakein. By applying transformations that convert inadmissible sequences into their admissible forms, we explain Lakein's counterexample from an algorithmic perspective. We provide additional counterexamples and prove that there exists an uncountably infinite family of counterexamples.
Complex Powers Of I Satisfying The Continued Fraction Functional Equation Over The Gaussian Integers, Matthew Niemiro '20
Complex Powers Of I Satisfying The Continued Fraction Functional Equation Over The Gaussian Integers, Matthew Niemiro '20
Exemplary Student Work
We investigate and then state the conditions under which iz satisfies the simple continued fraction functional equation for real and then complex z over the Gaussian integers.
A Fundamental Unit Of O_K, Susana L. Munoz
A Fundamental Unit Of O_K, Susana L. Munoz
Electronic Theses, Projects, and Dissertations
In the classical case we make use of Pells equation to compute units in the ring OF. Consider the parallel to the classical case and the quadratic field extension that creates the ring OK. We use the generalized Pell's equation to find the units in this ring since they are solutions. Through the use of continued fractions we may further characterize this ring and compute its units.
Some Contributions To The Sociology Of Numbers, Robert Dawson
Some Contributions To The Sociology Of Numbers, Robert Dawson
Journal of Humanistic Mathematics
Those who work with numbers eventually realize that they all have different personalities (the word "numbers" can of course be replaced by any number of other nouns here.) Here is one view of the issue.
Solutions To A Generalized Pell Equation, Kyle Christopher Castro
Solutions To A Generalized Pell Equation, Kyle Christopher Castro
Theses Digitization Project
This study aims to extend the notion of continued fractions to a new field Q (x)*, in order to find solutions to generalized Pell's Equations in Q [x] . The investigation of these new solutions to Pell's Equation will begin with the necessary extensions of theorems as they apply to polynomials with rational coefficients and fractions of such polynomials in order to describe each "family" of solutions.
Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su
Convergence Of Random Walks On The Circle Generated By An Irrational Rotation, Francis E. Su
All HMC Faculty Publications and Research
Fix . Consider the random walk on the circle which proceeds by repeatedly rotating points forward or backward, with probability , by an angle . This paper analyzes the rate of convergence of this walk to the uniform distribution under ``discrepancy'' distance. The rate depends on the continued fraction properties of the number . We obtain bounds for rates when is any irrational, and a sharp rate when is a quadratic irrational. In that case the discrepancy falls as (up to constant factors), where is the number of steps in the walk. This is the first example of a sharp …