More numerical evidence on the uniqueness of Markov Numbers

A Markov triple is a solution in postive integers of the equation x²+y²+z²=3 xyz. The maximum of the triple is a Markov Number. It is conjectured (Cassels "Introduction to Diophantine Approximations") that 2 distinct Markov triples cannot share the same Markov Number. Rosen & Patterson (Math. of Comp. 25 (1971)) tested the conjecture up to 10³⁰ by direct computation of the Markov Numbers. Modular Arithmetic was used here to carry the computation up to 10¹⁰⁵. No duplication was found. In addition the number of Markov numbers with N or less decimal digits is found to be approximately N² as in Rosen & Patterson. 1975 BIT Foundations.

Overview