#### Document Type

Article

#### Publication Date

Spring 2023

#### First Advisor

Joshua Holden

#### Abstract

This thesis is an exploration of Quantum Computing applied to Pell’s equation in an attempt to find solutions to the Billiard Ball Problem. Pell’s equation is a Diophantine equation in the form of x^{2} − ny^{2} = 1, where n is a given positive nonsquare integer, and integer solutions are sought for x and y. We will be applying Hallgren’s algorithm for finding irrational periods in functions, in the context of billiard balls and their movement on a friction-less unit square billiard table. Our central research question has been the following: Given the cutting sequence of the billiard ball’s movement, can you find the irrational slope value in which the billiard ball was put in motion?

#### Recommended Citation

Choi, Sangheon, "Applying Hallgren’s algorithm for solving Pell’s equation to finding the irrational slope of the launch of a billiard ball" (2023). *Mathematical Sciences Technical Reports (MSTR)*. 184.

https://scholar.rose-hulman.edu/math_mstr/184