Inaug.-diss.--Universität Zürich. / Lebenslauf. Bibliographical foot-notes.

Thesis (Ph. D.)--University of Texas at Austin, 1999. / Vita. Includes bibliographical references (leaves 20-22). Available also in a digital version from Dissertation Abstracts.

Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas. / Science, Faculty of / Mathematics, Department of / Graduate

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What constitutes an adequate solution to a mathematical problem? When is an adequate solution a 'good' solution? In this thesis I consider these questions in relation to two Diophantine equations, namely, x$ sp2$ + k = y$ sp3$ and 6y$ sp2$ = x(x + 1)(2x + 1). The first dates back to Diophantus himself (c. 250 AD) while the second can be traced to a puzzle proposed by Edouard Lucas in 1875. Each of these equations has attracted a number of solutions and each solution reveals something about its era. An examination and comparison of these solutions will give us an opportunity to reflect on some of the criteria used for judging proofs in mathematics. In particular, we shall see that contemporary computer technology has made a certain kind of solution to these equations acceptable which might have seemed pointless, incomplete or inelegant to the mathematicians who first studied them. Included among these 'computer solutions' is my own solution to 6y$ sp2$ = x(x + 1)(2x + 1).

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Thesis (doctoral)--Universitate Litteraria Friderica Guilelma, 1835. / Vita.

Thesis (M.S.)--Boise State University, 2009. / Includes abstract. Includes bibliographical references (leaf 82).