{"id":{"repo_id":"lethbridge","oai_identifier":"oai:opus.uleth.ca:10133/6591"},"canonical_url":"https://search.dev.ndltd.org/etd/lethbridge/oai:opus.uleth.ca:10133/6591","repository":{"repo_id":"lethbridge","name":"University of Lethbridge","base_url":"https://opus.uleth.ca/server/oai/request"},"display":{"title":"On the quality of the ABC-solutions","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.","abstract_html":"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.","abstract_has_math":false,"creators":["Bolvardizadeh, Solaleh","University of Lethbridge. Faculty of Arts and Science"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2023,"date_issued":"2023","date_published":"2023","updated_at":"2026-07-27T20:02:29Z","subjects":["ABC-conjecture","ABC-solutions","Lucas sequences","Diophantine equations","triple of integers"],"languages":[],"rights":[],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:10133/6591"],"render_values":[{"text":"hdl:10133/6591","href":null,"code":true}]}]},"links":{"outbound_url":null,"outbound_label":null,"outbound_source":null},"metadata_groups":[{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2023"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["ABC-conjecture","ABC-solutions","Lucas sequences","Diophantine equations","triple of integers"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["hdl:10133/6591"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.other","label":"Dc Description Other","values":["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."]},{"key":"dc:title","label":"Title","values":["On the quality of the ABC-solutions"]}]}],"canonical_facts":{"dc:date.issued":["2023"],"dc:description.other":["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."],"dc:identifier":["hdl:10133/6591"],"dc:subject":["ABC-conjecture","ABC-solutions","Lucas sequences","Diophantine equations","triple of integers"],"dc:title":["On the quality of the ABC-solutions"],"dc:type":["Thesis"]},"updated_at":"2026-07-27T20:02:29Z"}