Pell’s Equations Through Dynamical Systems

Abstract

Much work in number theory has been done since the 18th century on finding integer solutions to equations of the form x² − Dy² = 1, where D is a non-square integer. These equations, called Pells equations, have been shown to have infinite number of solutions that require finding the first non-trivial solution. There is no simple relationship between the parameter, D, and the size of its smallest non- trivial solution. In the late 18th century, Euler generated an iterative method to compute non-trivial solutions. In this work Eulers method is reduced to a function of three integers that can be iterated. Using more modern tools of dynamical systems, it is possible to explore the dynamics of this map as well as the stopping time of the algorithm. The stopping time will give an indication of the size of the smallest non-trivial solution. The developments of this research and the increased understanding of Pells equations allows people to solve and work with these and other similar equations that are used on a daily basis in many different disciplines of study.