On the number of Markoff numbers below a given bound

Author:
Don Zagier

Journal:
Math. Comp. **39** (1982), 709-723

MSC:
Primary 10F20; Secondary 10A20, 10B10

DOI:
https://doi.org/10.1090/S0025-5718-1982-0669663-7

MathSciNet review:
669663

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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 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.

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Article copyright:
© Copyright 1982
American Mathematical Society