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The asymptotic growth of integer solutions to the Rosenberger equations

Published online by Cambridge University Press:  17 April 2009

Arthur Baragar  and
Kensaku Umeda
Arthur Baragar
Affiliation:
Department of Mathematical Sciences, University of Nevada Las Vegas, 4505 Maryland Parkway, Las Vegas, NV 89154–4020, United States of America, e-mail: [email protected]
Kensaku Umeda
Affiliation:
Department of Mathematical Sciences, Florida Atlantic University, Boca Raton, FL 33431–0991, United States of America, e-mail: [email protected]
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Zagier showed that the number of integer solutions to the Markoff equation with components bounded by T grows asymptotically like C(log T)2, where C is explicity computable. Rosenberger showed that there are only a finite number of equations ax2 + by2 + cz2 = dxyz with a, b, and c dividing d, and for which the equation admits an infinite number of integer solutions. In this paper, we generalise Zagier's techniques so that they may be applied to the Rosenberger equations. We also apply these techniques to the equations ax2 + by2 + cz2 = dxyz + 1.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 69 , Issue 3 , June 2004 , pp. 481 - 497
Copyright
Copyright © Australian Mathematical Society 2004

References

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