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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.78.

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On the number of Markoff numbers below a given bound
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by Don Zagier PDF
Math. Comp. 39 (1982), 709-723 Request permission

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.
References
  • 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 10.1007/BF01419427
  • 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, Mathematics Lecture Series, No. 3, Publish or Perish, Inc., Boston, Mass., 1974. MR 0650832
  • A. Markoff, Sur les formes quadratiques binaires indéfinies, Math. Ann. 17 (1880), no. 3, 379–399 (French). MR 1510073, DOI 10.1007/BF01446234
  • 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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Additional Information
  • © Copyright 1982 American Mathematical Society
  • 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