On the quality of the ABC-solutions

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Bolvardizadeh, Solaleh
University of Lethbridge. Faculty of Arts and Science
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Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science
An 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.
ABC-conjecture , ABC-solutions , Lucas sequences , Diophantine equations , triple of integers