uniqueness conjecture for Markov numbers

Conjecture. Given a Markov numberMathworldPlanetmath z>1, there are several other Markov numbers x and y such that x2+y2+z2=3xyz, but there is only set of values of x and y satisfying the inequality z>yx.

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-(x2+y2+25) we obtain the sequenceMathworldPlanetmath -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 pn and 2pn. Ying Zhang used these results to extend this to 4pn and 8pn.


  • 1 R. K. Guy, Unsolved Problems in Number TheoryMathworldPlanetmathPlanetmath New York: Springer-Verlag 2004: D12
  • 2 Ying Zhang, “CongruenceMathworldPlanetmathPlanetmathPlanetmath 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