Some numerical evidence concerning the uniqueness of the Markov numbers

A Markov triple is a set of three positive integers satisfying the diophantine equation $({x^2} + {y^2} + {z^2} = 3xyz)$. The maximum of the triple is called a Markov number. Although all Markov triples can be generated from the triple (1,1,1), it is not known whether it is possible to obtain $(p,{a_1},{b_1})$ and $(p,{a_2},{b_2})$, where p is the same Markov number for both triples. All Markov numbers not exceeding 30 digits were computed without turning up a duplication, lending some credence to the conjecture that the Markov numbers are unique.