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 . 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 and , 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.

**[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****[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****[5]**A. Markoff, ``Sur les formes quadratiques binaires indefinées,''*Math. Ann.*, v. 15, 1879, pp. 381-409.

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DOI:
http://dx.doi.org/10.1090/S0025-5718-1971-0300972-X

Keywords:
Markov number,
Markov triple,
diophantine equation,
binary tree,
node,
branch,
bit string,
preorder traversal algorithm

Article copyright:
© Copyright 1971
American Mathematical Society