2026-09-18T21:49:47.841119200Zhttps://opus.uleth.ca/server/oai/requestoai:opus.uleth.ca:10133/65912023-09-29T05:04:43Zcom_10133_3346com_10133_1com_10133_296col_10133_3347col_10133_298
ULethbridge OPUS Repository
supervisor
Akbary, Amir
author
Bolvardizadeh, Solaleh
author
University of Lethbridge. Faculty of Arts and Science
2023-09-28T15:44:45Z
2023-09-28T15:44:45Z
2023
https://hdl.handle.net/10133/6591
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.
en
ABC-conjecture
ABC-solutions
Lucas sequences
Diophantine equations
triple of integers
On the quality of the ABC-solutions
Thesis
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URL
http://localhost:4000/bitstreams/77f8e13b-4978-4b30-bcff-4364e26582e3/download
File
MD5
47458d2f7d4e812a039610db97c00577
548958
application/pdf
BOLVARDIZADEH_SOLALEH_MSC_2023.pdf
URL
http://localhost:4000/bitstreams/2503bfcc-e651-4a1f-8655-6bb5fb1d00ee/download
File
MD5
b63a4b4390058258b8d24bc975cd1291
104091
text/plain
BOLVARDIZADEH_SOLALEH_MSC_2023.pdf.txt