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
DOI:
https://doi.org/10.1090/S0025-5718-1971-0300972-X
MathSciNet review:
0300972
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Abstract | References | Similar Articles | Additional Information
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.
- 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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Additional Information
Keywords:
Markov number,
Markov triple,
diophantine equation,
binary tree,
node,
branch,
bit string,
preorder traversal algorithm
Article copyright:
© Copyright 1971
American Mathematical Society