{"id":{"repo_id":"unlv","oai_identifier":"oai:oasis.library.unlv.edu:rtds-2405"},"canonical_url":"https://search.dev.ndltd.org/etd/unlv/oai:oasis.library.unlv.edu:rtds-2405","repository":{"repo_id":"unlv","name":"University of Nevada - Las Vegas","base_url":"https://oasis.library.unlv.edu/do/oai/"},"display":{"title":"The asymptotic behavior of the integer solutions of the Rosenberger equations","abstract":"The Rosenberger equations are equations of the form: ax 2 + by2 + cz 2 = dxyz, where the sets of coefficients ( a, b, c, d) are all integers such that each of a, b, and c divides d, and the equations themselves have infinitely many integer solutions. Rosenberger has shown that there are only six such sets of coefficients, one of which is the Markoff equation, x2 + y2 + z2 = 3xyz. Zagier investigated the asymptotic behavior of the integer solutions of the Markoff equation. In this paper, we apply Zagier's techniques to the Rosenberger equations and show that the number N(T) of positive integer solutions that are bounded by T is N(T) = C(log T) 2 + O(log T(log log T) 2), where C is an explicitly computable constant that depends on the equation.","abstract_html":"The Rosenberger equations are equations of the form: ax 2 + by2 + cz 2 = dxyz, where the sets of coefficients ( a, b, c, d) are all integers such that each of a, b, and c divides d, and the equations themselves have infinitely many integer solutions. Rosenberger has shown that there are only six such sets of coefficients, one of which is the Markoff equation, x2 + y2 + z2 = 3xyz. Zagier investigated the asymptotic behavior of the integer solutions of the Markoff equation. In this paper, we apply Zagier's techniques to the Rosenberger equations and show that the number N(T) of positive integer solutions that are bounded by T is N(T) = C(log T) 2 + O(log T(log log T) 2), where C is an explicitly computable constant that depends on the equation.","abstract_has_math":false,"creators":["Umeda, Kensaku"],"institution":"University of Nevada, Las Vegas","degree_name":"Master of Science (MS)","degree_level":"Thesis","degree_discipline":"Mathematical Sciences","degree_department":null,"school":null,"contributors":["Arthur Baragar"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2002,"date_issued":"2002-01-01T08:00:00Z","date_published":"2002-01-01T08:00:00Z","updated_at":"2026-07-24T05:25:33Z","subjects":[],"languages":["English"],"rights":["IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/"],"rights_urls":[],"identifier_entries":[{"key":"dc:identifier","label":"Identifier","values":["https://oasis.library.unlv.edu/rtds/1406"],"render_values":[{"text":"https://oasis.library.unlv.edu/rtds/1406","href":"https://oasis.library.unlv.edu/rtds/1406","code":true}]}]},"links":{"outbound_url":"https://doi.org/10.25669/60of-zhxs","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor","label":"Contributor","values":["Arthur Baragar"]},{"key":"dc:creator","label":"Author","values":["Umeda, Kensaku"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:publisher","label":"Institution","values":["University of Nevada, Las Vegas"]},{"key":"dc:type","label":"Dc Type","values":["Text"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematical Sciences"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Thesis"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science (MS)"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["English"]},{"key":"dc:rights","label":"Dc Rights","values":["IN COPYRIGHT. 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In this paper, we apply Zagier's techniques to the Rosenberger equations and show that the number N(T) of positive integer solutions that are bounded by T is N(T) = C(log T) 2 + O(log T(log log T) 2), where C is an explicitly computable constant that depends on the equation."]},{"key":"dc:format","label":"Dc Format","values":["pdf"]},{"key":"dc:title","label":"Title","values":["The asymptotic behavior of the integer solutions of the Rosenberger equations"]}]}],"canonical_facts":{"dc:contributor":["Arthur Baragar"],"dc:creator":["Umeda, Kensaku"],"dc:description.abstract":["The Rosenberger equations are equations of the form: ax 2 + by2 + cz 2 = dxyz, where the sets of coefficients ( a, b, c, d) are all integers such that each of a, b, and c divides d, and the equations themselves have infinitely many integer solutions. Rosenberger has shown that there are only six such sets of coefficients, one of which is the Markoff equation, x2 + y2 + z2 = 3xyz. Zagier investigated the asymptotic behavior of the integer solutions of the Markoff equation. In this paper, we apply Zagier's techniques to the Rosenberger equations and show that the number N(T) of positive integer solutions that are bounded by T is N(T) = C(log T) 2 + O(log T(log log T) 2), where C is an explicitly computable constant that depends on the equation."],"dc:format":["pdf"],"dc:identifier":["10.25669/60of-zhxs","https://oasis.library.unlv.edu/rtds/1406","https://oasis.library.unlv.edu/context/rtds/article/2405/viewcontent/uc.pdf"],"dc:language":["English"],"dc:publisher":["University of Nevada, Las Vegas"],"dc:rights":["IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/"],"dc:title":["The asymptotic behavior of the integer solutions of the Rosenberger equations"],"dc:type":["Text"],"thesis:degree_discipline":["Mathematical Sciences"],"thesis:degree_level":["Thesis"],"thesis:degree_name":["Master of Science (MS)"]},"updated_at":"2026-07-24T05:25:33Z"}