Pell's Equation: History, Methods, and Number TheoryShow full item record
|Pell's Equation: History, Methods, and Number Theory
|The title of the thesis will be "Pell's Equations: History, Methods and Number Theory." The thesis will comprise three parts, each corresponding to one of the topics mentioned in the title. The first part will discuss the history of Pell's Equation. Pell's Equation is an equation of the form x^2 - Dy^2 = 1, where x and y are variables in which integer solutions are sought and D is an integer. We will first take our look at the history of this equation at the time of Fermat. We will examine how he challenged his colleagues to solve certain specific polynomials, such as x^2 - 61y^2 = 1. The task was to find all the integer solutions. Over a hundred years later, Lagrange would solve this equation through the use of continued fractions. We will then go into how Pell's Equation has even a longer history in India. We will discuss how these problems were being discussed and solved as early as even the 6th century CE. In fact, the very equation x^2 - 61y^2 = 1 was solved by the Indian mathematician Bhâskara II. The methods used by the Indians mathematicians are described in this part as well, since they are no longer well known, and therefore of historical interest. We will conclude this part by describing how Pell's Equation was known even in the time of Archimedes. The so-called Cattle Problem would not be solved completely until the 1980s with the use of supercomputers. In the second part of this thesis the methods of solving Pell's equation will be discussed in detail. The basic concepts in the theory of continued fractions will be introduced. It is built upon the Euclidean algorithm. A complete description of the structure of all solutions to any Pell's equation will be given in this part. The specific equation x^2 - 61y^2 = 1 will be solved a second time using the continued fractions. A third method of solving Pell's equations, this time through an existence proof, will be discussed in the third part of the thesis. While an existence proof does not directly exhibit a solution, it can often throw light on the essence of a problem even more effectively than an algorithmic approach. The proof will be based on the idea of approximating irrational numbers by the rational ones. The relevant properties required will be introduced in conjunction with a careful study of the most essential parts of the theory of continued fractions. These properties will be proved in full. In the final section of this thesis, some of the many connections between Pell's equation and modern mathematics will be explored, though very briefly. The solutions of a Pell's equation are always infinite in number. It will be shown that they have an elegant structure known as a group in abstract algebra. Moreover, it will be shown that this group forms a part of algebraic number field. Thus, Pell's Equation turns out to be an essential part in the mathematical development of quadratic number fields. In conclusion, beyond the comparisons between the Indian methods and the modern concepts, and beyond the existence proof and the much wider and highly active field of Diophantine Approximations, the theory of continued fractions and the results of Lagrange can be seen to give rise to a broader, more philosophical question: is the notion of numbers and their classification purely dictated by what one may vaguely call "the truth," or can it perhaps be a mental construct, subject to, or even dictated by, our perception of "truth"?
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- Undergraduate Honors Papers