On the quality of the ABC-solutions

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dc.contributor.authorBolvardizadeh, Solaleh
dc.contributor.authorUniversity of Lethbridge. Faculty of Arts and Science
dc.contributor.supervisorAkbary, Amir
dc.date.accessioned2023-09-28T15:44:45Z
dc.date.available2023-09-28T15:44:45Z
dc.date.issued2023
dc.degree.levelMasters
dc.description.abstractAn ABC-solution is a triple (a,b,c) of integers such that gcd(a,b,c) = 1 and a+b = c. The quality of an ABC-solution is defined as q(a,b,c) = max{log |a|, log |b|, log |c|} / log rad(|abc|) , where rad(|abc|) is the product of distinct prime factors of |abc|. The ABC-conjecture states that given ε > 0 the number of the ABC-solutions (a,b,c) with q(a,b,c) ≥ 1+ε is finite. In this thesis, under the ABC-conjecture, we explore the quality of certain families of the ABC-solutions formed by terms in Lucas and associated Lucas sequences. We also unconditionally introduce a new family of ABC-solutions with quality > 1. In addition, we provide an upper bound on the quality of the ABC-solutions assuming an explicit version of the ABC-conjecture proposed by Alan Baker. Assuming this explicit upper bound for the quality, we explore the solutions to two Diophantine equations in integers, namely xn+yn = n!zn and n!+1 = m2. Next, inspired by the work of Pink and Szikszai, we provide the solutions to a generalization of n!+1 = m2 in Lucas sequences. Furthermore, we study the S-unit equations and find all the ABC-solutions with rad(ABC) = 30. Lastly, we consider another explicit version of the ABC-conjecture proposed by Baker and show that this conjecture is false by examining the newfound good ABC-solutions.
dc.identifier.urihttps://hdl.handle.net/10133/6591
dc.language.isoen
dc.proquestyesNo
dc.publisherLethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science
dc.publisher.departmentDepartment of Mathematics and Computer Science
dc.publisher.facultyArts and Science
dc.relation.ispartofseriesThesis (University of Lethbridge. Faculty of Arts and Science)
dc.subjectABC-conjecture
dc.subjectABC-solutions
dc.subjectLucas sequences
dc.subjectDiophantine equations
dc.subjecttriple of integers
dc.subject.lcshNumber theory
dc.subject.lcshLucas numbers
dc.subject.lcshDiophantine equations
dc.subject.lcshMathematics
dc.subject.lcshNumbers, Prime
dc.subject.lcshAlgebra
dc.subject.lcshDissertations, Academic
dc.titleOn the quality of the ABC-solutions
dc.typeThesis
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