Markov number

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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 in 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. ( 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. The largest number at any level is always about a third from the bottom. All the Markov numbers on the regions adjacent to 2's region are odd-indexed Pell numbers (or numbers n such that [2n^2 - 1] is a square, ), and all the Markov numbers on the regions adjacent to 1's region are odd-indexed Fibonacci numbers (). Thus, there are infinitely many Markov triples of the form

[(1, F_, F_),\,]

where F_x is the xth Fibonacci number. Likewise, there are infinitely many Markov triples of the form

[(1, P_, P_),\,]

where P_x is the xth Pell number.

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. It's not necessary that [x < y < z] in order for the [(x, y, 3xy - z)] to yield another triple. In fact, if one doesn't change the order of the members before applying the transform again, it returns the same triple one started with. Thus, starting with (1, 1, 2) and trading y and z before each iteration of the transform lists Markov triples with Fibonacci numbers. Starting with that same triplet and trading x and z before each iteration gives the triples with Pell numbers.

The nth Lagrange number can be calculated from the nth Markov number with the formula

[L_n = \sqrt^2}}.\,]

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 preferable because "Markoff number" might be misunderstood as "mark-off number."

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