Approximating Markov’s equation

Markov numbers are integer solutions to x² + y² + z² = 3xyz. The Wikipedia article on Markov numbers mentions that Don Zagier studied Markov numbers by looking the approximating equation x² + y² + z² = 3xyz + 4/9 which is equivalent to f(x) + f(y) = f(z) where f(t) is defined as arccosh(3t/2). It wasn’t clear to me why the two previous equations are equivalent, so I’m writing this post to show that they are equivalent. Examples Before showing the equivalence of Zagier’s two equations, let’s look at an example that shows solutions to his second equation approximate solutions to Markov’s equation. The following code verifies that (5, 13, 194) is a solution to Markov’s equation. x, y, z = 5, 13, 194 assert(x**2 + y**2 + z**2 == 3*x*y*z) With the same x and y above, let’s show that the z in Zagier’s second equation is close to the z above. from math import cosh, acosh f = lambda t: acosh(3*t/2) g = lambda t: cosh(t)*2/3 z = g(f(x) + f(y)) print(z) This gives z = 194.0023, which is close…