UniquenessConjectureForMarkovNumbers 2007-07-28 17:08:52 2007-07-31 13:08:21 Conjecture Markov number % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here Conjecture. Given a Markov number $z > 1$, there are several other Markov numbers $x$ and $y$ such that $x^2 + y^2 + z^2 = 3xyz$, but there is only set of values of $x$ and $y$ satisfying the inequality $z > y \geq x$. The conjecture is easy enough to check for small values. For example, for $z = 5$, we could even test $x$ and $y$ we know not to be Markov numbers with minimum increase in computational overhead. Trying the triples in order: (1, 1, 5), (1, 2, 5), (1, 3, 5), (1, 4, 5), (2, 1, 5), ... (4, 4, 5) against $15xy - (x^2 + y^2 + 25)$ we obtain the sequence $-12$, 0, 10, 18, 0, 27, 52, 75, 10, 52, 92, 130, 18, 75, 130, 183. It doesn't take significantly larger Markov numbers to show the need for a general proof of uniqueness. Many attempted proofs have been submitted, but Richard Guy dismisses them all as seemingly faulty. A divide-and-conquer approach to the problem has yielded encouraging results, however. Baragar proved the uniqueness of prime Markov numbers $p$ as well as semiprimes $2p$. Schmutz then proved the uniqueness of Markov numbers of the forms $p^n$ and $2p^n$. Ying Zhang used these results to extend this to $4p^n$ and $8p^n$. \begin{thebibliography}{2} \bibitem{rg} R. K. Guy, {\it Unsolved Problems in Number Theory} New York: Springer-Verlag 2004: D12 \bibitem{yz} Ying Zhang, ``Congruence and Uniqueness of Certain Markov Numbers'' {\it Acta Arithmetica} {\bf 128} 3 (2007): 297 \end{thebibliography}