The uniqueness of the Markoff numbers

Rosenberger, Gerhard

doi:10.1090/S0025-5718-1976-0396413-1

The uniqueness of the Markoff numbers

Author: Gerhard Rosenberger
Journal: Math. Comp. 30 (1976), 361-365
MSC: Primary 10C30; Secondary 10B10
DOI: https://doi.org/10.1090/S0025-5718-1976-0396413-1
MathSciNet review: 0396413
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Abstract: A Markoff triple is a set of three positive integers satisfying the diophantine equation ${x^2} + {y^2} + {z^2} = 3xyz$ . The maximum of the three numbers is called a Markoff number. We show: If there are Markoff triples $({x_1},{y_1},z)$ and $({x_2},{y_2},z)$ with the same Markoff number z, then ${x_1} = {x_2}$ or ${x_1} = {y_2}$ .

[1] I. BOROSH, "Numerical evidence on the uniqueness of Markoff numbers," Notices Amer. Math. Soc., v. 21, 1974, p. A-55. Abstract #711-10-32.
[2] Harvey Cohn, Markoff forms and primitive words, Math. Ann. 196 (1972), 8–22. MR 297847, https://doi.org/10.1007/BF01419427
[3] J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1917), no. 1, 385–397 (German). MR 1511907, https://doi.org/10.1007/BF01457113
[4] D. Rosen and G. S. Patterson Jr., Some numerical evidence concerning the uniqueness of the Markov numbers, Math. Comp. 25 (1971), 919–921. MR 300972, https://doi.org/10.1090/S0025-5718-1971-0300972-X
[5] Gerhard Rosenberger, Fuchssche Gruppen, die freies Produkt zweier zyklischer Gruppen sind, und die Gleichung 𝑥²+𝑦²+𝑧²=𝑥𝑦𝑧, Math. Ann. 199 (1972), 213–227 (German). MR 340447, https://doi.org/10.1007/BF01429875