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Please use this identifier to cite or link to this item: http://hdl.handle.net/1974/7062

Title: Generalization of Ruderman's Problem to Imaginary Quadratic Fields
Authors: Rundle, Robert John

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Keywords: exponential Diophantine equations
Number Theory
abc conjecture
Schmidt's Subspace theorem
Diophantine Equations
Issue Date: 13-Apr-2012
Series/Report no.: Canadian theses
Abstract: In 1974, H. Ruderman posed the following question: If $(2^m-2^n)|(3^m-3^n)$, then does it follow that $(2^m-2^n)|(x^m-x^n)$ for every integer $x$? This problem is still open. However, in 2011, M. R. Murty and V. K. Murty showed that there are only finitely many $(m,n)$ for which the hypothesis holds. In this thesis, we examine two generalizations of this problem. The first is replacing 2 and 3 with arbitrary integers $a$ and $b$. The second is to replace 2 and 3 with arbitrary algebraic integers from an imaginary quadratic field. In both of these cases we have shown that there are only finitely many $(m,n)$ for which the hypothesis holds. To get the second result we also generalized a result by Bugeaud, Corvaja and Zannier from the integers to imaginary quadratic fields. In the last half of the thesis we use the abc conjecture and some related conjectures to study some exponential Diophantine equations. We study the Pillai conjecture and the Erd\"{o}s-Woods conjecture and show that they are implied by the abc conjecture and that when we use an effective version, very clean bounds for the conjectures are implied.
Description: Thesis (Ph.D, Mathematics & Statistics) -- Queen's University, 2012-04-13 12:04:14.252
URI: http://hdl.handle.net/1974/7062
Appears in Collections:Queen's Graduate Theses and Dissertations
Department of Mathematics and Statistics Graduate Theses

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