## Investigations on some exponential congruences

 dc.contributor.author Bose, Arnab dc.contributor.author University of Lethbridge. Faculty of Arts and Science dc.contributor.supervisor Akbary-Majdabadno, Amir dc.date.accessioned 2016-08-05T17:21:14Z dc.date.available 2016-08-05T17:21:14Z dc.date.issued 2016 dc.degree.level Masters en_US dc.description.abstract Selfridge asked for what positive integers a and b with a > b, does 2a — 2b divide na — nb for all n e N. The problem was solved by various people who showed that the above problem is true only for (a, b) e S, where S = {(2,1), (3,1), (4,2), (5,1), (5,3), (6,2), (7,3), (8,2), (8,4), (9,3), (14,2), (15,3), (16,4)}. In this thesis, we prove two generalizations of the above problem. Theorem. For a fixed positive integer m, na — nb = 0 (mod ma — mb) has a solution in (a, b) e N2 with a > b, for all integers n > m if and only ifm = 2 and (a, b) e S, where S is as given above. Zaharescu and Vajaitu considered a generalization of Selfridge’s problem in algebraic number fields. Our second result makes their theorem explicit and provides explicit bounds for the solutions. Next, we give a conditional resolution to a problem proposed by Ruderman which is related to Selfridge’s problem and also investigate some generalizations. Lastly, we use a particular case of the Schmidt Subspace Theorem and generalize a result proved by Bugeaud, Corvaja and Zannier [2]. en_US dc.embargo No en_US dc.identifier.uri https://hdl.handle.net/10133/4567 dc.language.iso en_CA en_US dc.proquest.subject 0405 en_US dc.proquestyes Yes en_US dc.publisher Lethbridge, Alta : University of Lethbridge, Dept. of Mathematics and Computer Science en_US dc.publisher.department Department of Mathematics and Computer Science en_US dc.publisher.faculty Arts and Science en_US dc.relation.ispartofseries Thesis (University of Lethbridge. Faculty of Arts and Science) en_US dc.subject mathematics en_US dc.subject number theory en_US dc.subject Ruderman's problem en_US dc.subject Schmidt Subspace Theorem en_US dc.subject Selfridge's problem en_US dc.title Investigations on some exponential congruences en_US dc.type Thesis en_US
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