A Markov number is one of the values \(x\), \(y\), or \(z\) in a positive-integer solution of the Markov equation \(x^2 + y^2 + z^2 = 3xyz\). Starting from the solution \((1,1,1)\), every other solution is reached by "Vieta jumping" — fixing two values and flipping the third — which arranges all solutions into an infinite binary tree. The numbers that appear: 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985.
Many Markov numbers are Fibonacci numbers (1, 2, 5, 13, 34, 89, 233) or Pell numbers (1, 2, 5, 29, 169, 985) — the two edges of the Markov tree. The long-standing unicity conjecture asks whether each Markov number appears as the largest element of exactly one triple; it remains open. Markov numbers also govern how badly certain irrational numbers can be approximated by rationals (the Markov spectrum), named for Andrey Markov.