Approximating Markov’s equation
Markov numbers are integer solutions to x² + y² + z² = 3xyz. Don Zagier studied them using the approximating equation x² + y² + z² = 3xyz + 4/9, which is equivalent to f(x) + f(y) = f(z) where f(t) = arccosh(3t/2).
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Don Zagier discovered that the Markov equation x² + y² + z² = 3xyz can be approximated by a much simpler equation, offering a new perspective on the classical Diophantine problem. His approximation reveals a deep connection between number theory and hyperbolic geometry.