Markov number

A Markov number or Markoff number is an integer x, y or z that is part of a solution to the Markov Diophantine equation

$x^{2}+y^{2}+z^{2}=3xyz$

The first few Markov numbers are

1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985, 1325, ... (sequence A002559 in the OEIS)

appearing in the solutions

{1, 1, 1}, {1, 1, 2}, {1, 2, 5}, {1, 5, 13}, {2, 5, 29}, {1, 13, 34}, {1, 34, 89}, {2, 29, 169}, {5, 13, 194}, {1, 89, 233}, {5, 29, 433}, {89, 233, 610}, etc.

There are infinitely many Markov numbers and Markov triples. Any Markov number appears in at least three solutions, but is the largest integer in only one solution. Knowing one Markov triple ${x,y,z}$ one can find another Markov triple, of the form ${x,y,3xy-z}$. Markov numbers are not always prime, but members of a Markov triple are always coprime (with the exception of the first two triples). (OEIS: A030452 lists Markov numbers that appear in solutions where one of the other two terms is 5).

Due to the commutative properties of addition and multiplication, the solutions may be arranged in any order, but it might be helpful to arrange each Markov triple in ascending order, and the triples in order by highest integer contained (as above).

The Markov numbers can also be arranged in a binary tree. Thus, all the Markov numbers on the leftmost branches are odd-indexed Fibonacci numbers (OEIS: A001519), and all the Markov numbers on the rightmost branches are numbers n such that $2n^{2}-1$ is a square (OEIS: A001653). The largest number at any level is always about a third from the right.

Markov numbers are named after the Russian mathematician Andrey Markov. Due to the different but equally valid ways of transliterating Cyrillic, the term is written as "Markoff numbers" in some literature. But in this particular case, "Markov" might be preferrable because "Markoff number" might be misunderstood as "mark-off number."