Diophantine equation, is equivalent to with f(t) = arcosh(3t&hairsp;&hairsp;/&hairsp;2). The conjecture was proved by Greg McShane and Igor Rivin in 1995 using techniques from hyperbolic geometry. The nth Lagrange number can be calculated from the nth Markov number with the formula The Markov numbers are sums of (non-unique) pairs of squares. Markov's theorem showed that if is an indefinite binary quadratic form with real coefficients and discriminant , then there are integers x, y for which f takes a nonzero value of absolute value at most unless f is a Markov form: a constant times a form such that where (p, q, r) is a Markov triple. Matrices Let Tr denote the trace function over matrices. If X and Y are in SL2(ℂ), then Tr(X) Tr(Y) Tr(X⋅Y) + Tr(X⋅Y⋅X −1⋅Y −1) + 2 = Tr(X)2 + Tr(Y)2 + Tr(X⋅Y)2 so that if Tr(X⋅Y⋅X −1⋅Y −1) = −2 then Tr(X) Tr(Y) Tr(X⋅Y) = Tr(X)2 + Tr(Y)2 + Tr(X⋅Y)2 In particular if X and Y also have integer entries then Tr(X)/3, Tr(Y)/3, and Tr(X⋅Y)/3 are a Markov triple. If X⋅Y⋅Z = I then Tr(X⋅Y) = Tr(Z), so more symmetrically if X, Y, and Z are in SL2(ℤ) with X⋅Y⋅Z = I and the commutator of two of them has trace −2, then their traces/3 are a Markov triple. See also Markov spectrum Notes References Diophantine equations Diophantine approximation Fibonacci numbers