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

 

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?)

  • [1] 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 (19,396h)
  • [2] L. E. Dickson, Studies in the Theory of Numbers, Univ. of Chicago Press, Chicago, Ill., 1930.
  • [3] G. Frobenius, ``Über die Markoffschen Zahlen,'' S.-B. Preuss. Akad. Wiss., v. 1913, pp. 458-487.
  • [4] 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 (51 #14624)
  • [5] A. Markoff, ``Sur les formes quadratiques binaires indefinées,'' Math. Ann., v. 15, 1879, pp. 381-409.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0025-5718-1971-0300972-X
PII: S 0025-5718(1971)0300972-X
Keywords: Markov number, Markov triple, diophantine equation, binary tree, node, branch, bit string, preorder traversal algorithm
Article copyright: © Copyright 1971 American Mathematical Society