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Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)


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
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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PII: S 0025-5718(1982)0669663-7
Article copyright: © Copyright 1982 American Mathematical Society