# Bounds on the period of the continued fraction after a M\"obius transformation

## TL;DR

This paper establishes explicit bounds on how the period of continued fractions of quadratic numbers changes under Möbius transformations, providing sharp bounds with illustrative examples.

## Contribution

It introduces new explicit bounds on the continued fraction period after Möbius transformations, enhancing understanding of their effect on quadratic numbers.

## Key findings

- Derived explicit upper and lower bounds for continued fraction periods
- Provided examples demonstrating the bounds are sharp
- Enhanced theoretical understanding of Möbius transformations on quadratic numbers

## Abstract

We study M\"obius transformations (also known as linear fractional transformations) of quadratic numbers. We construct explicit upper and lower bounds on the period of the continued fraction expansion of a transformed number as a function of the period of the continued fraction expansion of the original number. We provide examples that show that the bound is sharp.