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Some numerical evidence concerning the uniqueness of the Markov numbers

Authors: D. Rosen and G. S. Patterson
Journal: Math. Comp. 25 (1971), 919-921
MSC: Primary 10B10
MathSciNet review: 0300972
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Abstract: A Markov triple is a set of three positive integers satisfying the diophantine equation $({x^2} + {y^2} + {z^2} = 3xyz)$. The maximum of the triple is called a Markov number. Although all Markov triples can be generated from the triple (1,1,1), it is not known whether it is possible to obtain $(p,{a_1},{b_1})$ and $(p,{a_2},{b_2})$, where p is the same Markov number for both triples. All Markov numbers not exceeding 30 digits were computed without turning up a duplication, lending some credence to the conjecture that the Markov numbers are unique.

References [Enhancements On Off] (What's this?)

  • 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
  • L. E. Dickson, Studies in the Theory of Numbers, Univ. of Chicago Press, Chicago, Ill., 1930. G. Frobenius, “Über die Markoffschen Zahlen,” S.-B. Preuss. Akad. Wiss., v. 1913, pp. 458-487.
  • Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR 0378456
  • A. Markoff, “Sur les formes quadratiques binaires indefinées,” Math. Ann., v. 15, 1879, pp. 381-409.

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Keywords: Markov number, Markov triple, diophantine equation, binary tree, node, branch, bit string, preorder traversal algorithm
Article copyright: © Copyright 1971 American Mathematical Society