dc.description.abstract | This thesis is a comparative study between two approaches to a classical topic in number theory generally known as Pell's Equation. The equation has the general form
x2 - Ny2 = 1,
where N is a positive integer not equal to a perfect square. Only solutions in positive integers are sought. Because there are two unknowns, this is an example of an indeterminate equation; such equations are also called Diophantine equations, after Diophantus, the 3rd century Alexandrian mathematician whose masterpiece, Arithmetica, exerted an out-sized influence on the development of number theory.
The comparisons are temporal, geographical, as well as methodological. Diophantine equations have had a very long history. Though not written in the form we do now, the ancient Greeks, the Chinese, and the Indians were all interested in various genres of such equations. Each culture supplied its own motivation, however.
In the case of Pell's equations, the Indians, after mastering Diophantine equations of the linear type, considered them in the 7th century. Brahmagupta obtained foundational results concerning what he called "square-nature" problems, and singled out Pell's equations for the illustration of his general methods. Five centuries later, Bhaskaracharya completed the study of Pell's equation by establishing his chakravala, or cyclic method, which is totally general.
The Indian investigations of this topic, however, were entirely unknown to the Europeans for many centuries. When Fermat initiated his ground-breaking number theoretic studies in the 17th century, he thought he was recasting Diophantus's work on a grander foundation. Pell's Equation was a case study of the properties of what would become "units of algebraic integers"; but when Fermat challenged his contemporary peers to solve such equations, he probably imagined that no one without his insight into the bigger context would be successful.
William Brouncker, viscount and future President of the Royal Society, however, proved Fermat wrong by discovering a complete method within a few months. In retrospect, Lord Brouncker's discovery was an inadvertent illustration of how much the two seemingly distinct lines of research, the Indian and the European, had in common, when viewed methodologically. The kuttaka, or the pulverizer method, which both Brahmagupta and Bhaskaracharya relied upon, was in essence a variation of the familiar Euclidean Algorithm. Lord Brouncker, on the other hand, utilized the Euclidean Algorithm in an essential way as well; his investigation of what amounted to a study of continued fractions heavily depended on it. The main conclusion of this study is that the continued fractions expansion for the square root of N in the above equation is infinite but periodic. It was André Weil who first suggested that Brouncker's periodicity and Bhaskara's cyclicity are in fact closely related. This fact has been verified concretely in this thesis.
Moreover, this thesis also addresses the distinct styles in which the methods of solving Pell's equations are represented, in the Indian and Western literature respectively. The most glaring aspect of this stylistic distinction may be found in the specific techniques highlighted; while the Western style is to bring out the comprehensiveness and generality of the method, the Indian presentation emphasizes efficiency. The resulting apparent differences are so great that questions have been raised as to the completeness of the Indian method. (Chauvinism may also have played a role here.) It will be argued, though only briefly, that the differences are better understood as manifestations of cultural and philosophical influences at work.
"Comparative mathematics" is not a known academic discipline at the present time; rather, it is an emerging field of study. Only recently have the possibility and potential of systematic investigations of this type begun to be explored and explained. It is hoped that this thesis may be regarded as a sample specimen in support of such endeavors. | |