On the Number of Markoff Numbers Below a Given Bound
Don Bernard Zagier · Mathematics of Computation · 1982
According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than $\frac {1}{3}$ of the square root of the discriminant) are in 1 : 1 correspondence with the solutions of the Diophantine equation ${p^2} + {q^2} + {r^2} = 3pqr$. By relating Markoffs algorithm for finding solutions of this equation to a problem of counting lattice points in triangles, it is shown that the number of solutions less than x equals $C{\log ^2}3x + O(\log x\log {\log ^2}x)$ with an explicitly computable constant $C = 0.18071704711507 \ldots$ Numerical data up to ${10^{1300}}$ is presented which suggests that the true error term is considerably smaller.