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 $z>y\ge x$.
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 $2{p}^{n}$. Ying Zhang used these results to extend this to $4{p}^{n}$ and $8{p}^{n}$.
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 |
---|---|
Canonical name | UniquenessConjectureForMarkovNumbers |
Date of creation | 2013-03-22 17:26:16 |
Last modified on | 2013-03-22 17:26:16 |
Owner | PrimeFan (13766) |
Last modified by | PrimeFan (13766) |
Numerical id | 5 |
Author | PrimeFan (13766) |
Entry type | Conjecture |
Classification | msc 11J06 |
Synonym | unicity conjecture for Markov numbers |