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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
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 ${x_1} = {y_2}$.
-
I. BOROSH, "Numerical evidence on the uniqueness of Markoff numbers," Notices Amer. Math. Soc., v. 21, 1974, p. A-55. Abstract #711-10-32.
- Harvey Cohn, Markoff forms and primitive words, Math. Ann. 196 (1972), 8โ22. MR 297847, DOI https://doi.org/10.1007/BF01419427
- J. Nielsen, Die Isomorphismen der allgemeinen, unendlichen Gruppe mit zwei Erzeugenden, Math. Ann. 78 (1917), no. 1, 385โ397 (German). MR 1511907, DOI https://doi.org/10.1007/BF01457113
- D. Rosen and G. S. Patterson Jr., Some numerical evidence concerning the uniqueness of the Markov numbers, Math. Comp. 25 (1971), 919โ921. MR 300972, DOI https://doi.org/10.1090/S0025-5718-1971-0300972-X
- Gerhard Rosenberger, Fuchssche Gruppen, die freies Produkt zweier zyklischer Gruppen sind, und die Gleichung $x^{2}+y^{2}+z^{2}=xyz$, Math. Ann. 199 (1972), 213โ227 (German). MR 340447, DOI https://doi.org/10.1007/BF01429875
Retrieve articles in Mathematics of Computation with MSC: 10C30, 10B10
Retrieve articles in all journals with MSC: 10C30, 10B10
Additional Information
Article copyright:
© Copyright 1976
American Mathematical Society