uniqueness conjecture for Markov numbers

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 $8p^{n}$.

References

• 1 R. K. Guy, Unsolved Problems in Number Theory New York: Springer-Verlag 2004: D12
• 2 Ying Zhang, “Congruence and Uniqueness of Certain Markov Numbers” Acta Arithmetica 128 3 (2007): 297
Title uniqueness conjecture for Markov numbers UniquenessConjectureForMarkovNumbers 2013-03-22 17:26:16 2013-03-22 17:26:16 PrimeFan (13766) PrimeFan (13766) 5 PrimeFan (13766) Conjecture msc 11J06 unicity conjecture for Markov numbers