The asymptotic growth of integer solutions to the Rosenberger equations

Abstract

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.