Master of Science (MS)
First Committee Member
Number of Pages
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.
Asymptotic; Behavior; Equations; Integer; Rosenberger; Solutions
University of Nevada, Las Vegas
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Umeda, Kensaku, "The asymptotic behavior of the integer solutions of the Rosenberger equations" (2002). UNLV Retrospective Theses & Dissertations. 1406.