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On the Number of Markoff Numbers Below a Given Bound

Don Zagier
Mathematics of Computation
Vol. 39, No. 160 (Oct., 1982), pp. 709-723
DOI: 10.2307/2007348
Stable URL: http://www.jstor.org/stable/2007348
Page Count: 15
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On the Number of Markoff Numbers Below a Given Bound
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Abstract

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 Markoff's 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.

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