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

Keywords:  exponential Diophantine equations Number Theory abc conjecture Schmidt's Subspace theorem Mathematics Diophantine Equations 
Issue Date:  13Apr2012 
Series/Report no.:  Canadian theses 
Abstract:  In 1974, H. Ruderman posed the following question: If $(2^m2^n)(3^m3^n)$, then does it follow that $(2^m2^n)(x^mx^n)$ 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}sWoods 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, 20120413 12:04:14.252 
URI:  http://hdl.handle.net/1974/7062 
Appears in Collections:  Mathematics & Statistics Graduate Theses Queen's Theses & Dissertations

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