Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-24T16:33:12.407Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Unicity Conjecture for Markoff Numbers

Published online by Cambridge University Press:  20 November 2018

Arthur Baragar
Arthur Baragar*
Affiliation:
Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, N2L 3G1, e-mail:, [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In 1913 Frobenius conjectured that for any positive integer m, there exists at most one pair of integers (x, y) with 0 ≤ x ≤ y ≤ m such that (x, y, m) is a solution to the Markoff equation: x2 + y2 + m2 = 3xym. We show this is true if either m, 3m — 2 or 3m + 2 is prime, twice a prime or four times a prime.

Keywords

11D25
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 1 , 01 March 1996 , pp. 3 - 9
Copyright
Copyright © Canadian Mathematical Society 1995

References

[B] Baragar, A., The Hurwitz Equations, Number Theory with an Emphasis on the Markoff Spectrum, Lecture Notes in Pure and Applied Mathematics, (A. Pollington and W. Moran), Marcel Dekker, New York 147(1993), 1–8.Google Scholar
[C] Cassels, J. W. S., An Introduction to Diophantine Approximation, Chapter II, Cambridge Univ. Press, Cambridge, 1957.Google Scholar
[F] Frobenius, G., Über die Markoffschen Zahlen, Akad. Wiss. Sitzungaber, (1913), 458–493.Google Scholar
[G] Guy, R., Unsolved Problems in Number Theory, New York, (1981).Google Scholar
[M] Markoff, A. A., Sur les formes binaires indéfinies, Math. Ann. 17(1880), 379–399.Google Scholar
[N-Z] Niven, I. and Zuckerman, H., An Introduction to the Theory of Numbers, John Wiley & Sons, New York, 1980.Google Scholar
[R-P] Rosen, D. and Patterson, G., Jr., Some Numerical Evidence Concerning the Uniqueness of the Markov Numbers Math. Comp. 25(1971), 919–921.Google Scholar
[R] Rosenberger, G., The Uniqueness of the Markoff Numbers, but see MR 53 #280, Math. Comp. 30(1976), 361–365.Google Scholar
[Z] Zagier, D., On the Number of Markoff Numbers Below a Given Bound, Math. Comp. 39(1982), 704–723.Google Scholar