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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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  • Alan Baker, Transcendental number theory, Cambridge University Press, London-New York, 1975. MR 0422171
  • J. W. S. Cassels, An introduction to Diophantine approximation, Cambridge Tracts in Mathematics and Mathematical Physics, No. 45, Cambridge University Press, New York, 1957. MR 0087708
  • Harvey Cohn, Markoff forms and primitive words, Math. Ann. 196 (1972), 8–22. MR 297847, DOI
  • Harvey Cohn, Minimal geodesics on Fricke’s torus-covering, Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978) Ann. of Math. Stud., vol. 97, Princeton Univ. Press, Princeton, N.J., 1981, pp. 73–85. MR 624806
  • Harvey Cohn, Growth types of Fibonacci and Markoff, Fibonacci Quart. 17 (1979), no. 2, 178–183. MR 536967
  • G. Frobenius, Über die Markoffschen Zahlen, Preuss. Akad. Wiss. Sitzungsberichte, 1913, pp. 458-487. C. Gurwood, Diophantine Approximation and the Markoff Chain, Thesis, New York University, 1976, Section VI. G. H. Hardy & J. E. Littlewood, "Some problems of Diophantine approximation: The lattice points of a right-angled triangle," (1st memoir), Proc. London Math. Soc. (2), v. 20, 1922, pp. 15-36, (2nd memoir), Abh. Math. Sem. Univ. Hamburg, v. 1, 1921, pp. 212-249. In Collected Papers of G. H. Hardy, Vol. I, pp. 136-158, 159-196, Clarendon Press, Oxford, 1966.
  • F. Hirzebruch and D. Zagier, The Atiyah-Singer theorem and elementary number theory, Publish or Perish, Inc., Boston, Mass., 1974. Mathematics Lecture Series, No. 3. MR 0650832
  • A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann. 17 (1880), no. 3, 379–399 (French). MR 1510073, DOI
  • Arnold Walfisz, Weylsche Exponentialsummen in der neueren Zahlentheorie, Mathematische Forschungsberichte, XV, VEB Deutscher Verlag der Wissenschaften, Berlin, 1963 (German). MR 0220685

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Article copyright: © Copyright 1982 American Mathematical Society