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Abstract

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.

Author Bio

Holly VanLooy is from Boise, Idaho and graduated from Boise State University in May 2024 with a B.S. in mathematics. She is now a Ph.D. student at the University of Utah. Her current mathematical interests are in differential equations and their applications to physics-based problems. When not doing math, VanLooy enjoys singing in choir and reading.

Carolina A. Rizzi graduated from Texas A&M International University in December 2023 with a B.S. in mathematics. Now, Rizzi is determined to complete her teacher and actuary credentials, and ultimately pursue a Mathematics Ph.D. with a specialization in Number Theory.

In the summer of 2023, the authors engaged in a mathematics REU at UT Tyler, focusing on complex continued fractions.

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