Asymptotic growth of Markoff-Hurwitz numbers

Arthur Baragar

Compositio Mathematica (1994)

  • Volume: 94, Issue: 1, page 1-18
  • ISSN: 0010-437X

How to cite


Baragar, Arthur. "Asymptotic growth of Markoff-Hurwitz numbers." Compositio Mathematica 94.1 (1994): 1-18. <>.

author = {Baragar, Arthur},
journal = {Compositio Mathematica},
keywords = {Markoff numbers; growth of integral solutions; symmetric equations of Hurwitz; multi-branched Euclidean trees; asymptotic growth},
language = {eng},
number = {1},
pages = {1-18},
publisher = {Kluwer Academic Publishers},
title = {Asymptotic growth of Markoff-Hurwitz numbers},
url = {},
volume = {94},
year = {1994},

AU - Baragar, Arthur
TI - Asymptotic growth of Markoff-Hurwitz numbers
JO - Compositio Mathematica
PY - 1994
PB - Kluwer Academic Publishers
VL - 94
IS - 1
SP - 1
EP - 18
LA - eng
KW - Markoff numbers; growth of integral solutions; symmetric equations of Hurwitz; multi-branched Euclidean trees; asymptotic growth
UR -
ER -


  1. [B1] A. Baragar: The Markoff Equation and Equations of Hurwitz, Ph.D. Thesis, Brown University, 1991. 
  2. [B2] A. Baragar: Integral solutions of Markoff-Hurwitz equations, J. Number Theory (to appear). Zbl0820.11016MR1295950
  3. [Ca] J.W.S. Cassels: An Introduction to Diophantine Approximation, Chap. II, Cambridge Univ. Press, Cambridge, 1957. Zbl0077.04801MR87708
  4. [Co] H. Cohn: Growth types of Fibonacci and Markoff, Fibonacci Quart.17(2) (1979) 178-183. Zbl0415.05018MR536967
  5. [He] N.P. Herzberg: On a problem of Hurwitz, Pac. J. Math.50 (1974) 485-493. Zbl0247.10010MR347731
  6. [Hu] A. Hurwitz: Über eine aufgabe der unmbestimmten analysis, Archiv. Math. Phys.3 (1907) 185-196;Mathematisch Werke, Vol. 2, Chap. LXX (1933 and 1962) 410-421. JFM38.0246.01
  7. [Ma] A.A. Markoff: Sur les formes binaires indéfinies, Math. Ann.17 (1880) 379-399. MR1510073JFM12.0143.02
  8. [Mo1] L.J. Mordell: On the integer solutions of the equation x2 + y2 + z2 + 2xyz = n, J. London Math. Soc.28 (1953) 500-510. Zbl0051.27802MR56619
  9. [Mo2] L.J. Mordell: Diophantine Equations, Chap. 13.4. Academic Press, New York, London, 1969, pp. 106-110. Zbl0188.34503MR249355
  10. [S] J.H. Silverman: The Markoff equation x2 + y2 + z2 = axyz over quadratic imaginary fields, J. Number Theory35(1) (1990) 72-104. Zbl0702.11012MR1054560
  11. [Z] D. Zagier: On the number of Markoff numbers below a given bound, Math. Comp.39 (1982) 709-723. Zbl0501.10015MR669663

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