On the quality of the ABC-solutions
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Date
2023
Authors
Bolvardizadeh, Solaleh
University of Lethbridge. Faculty of Arts and Science
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Publisher
Lethbridge, Alta. : University of Lethbridge, Dept. of Mathematics and Computer Science
Abstract
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.
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Keywords
ABC-conjecture , ABC-solutions , Lucas sequences , Diophantine equations , triple of integers