Applying Hallgren’s algorithm for solving Pell’s equation to finding the irrational slope of the launch of a billiard ball

Authors

Sangheon Choi, Rose-Hulman Institute of TechnologyFollow

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 x² − ny² = 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