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 $8p^n$ .

Bibliography

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