The asymptotic behavior of the integer solutions of the Rosenberger equations

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.

Name thesis:degree_name

Master of Science (MS)

Level thesis:degree_level

Thesis

Discipline thesis:degree_discipline

Mathematical Sciences

Grantor dc:publisher

University of Nevada, Las Vegas

Year

2002

Author dc:creator

• Umeda, Kensaku

Contributors dc:contributor

• Arthur Baragar

Statement dc:rights

• IN COPYRIGHT. For more information about this rights statement, please visit http://rightsstatements.org/vocab/InC/1.0/

Language dc:language

English

Identifier

https://oasis.library.unlv.edu/rtds/1406

OAI identifier oai:identifier

oai:oasis.library.unlv.edu:rtds-2405