On the number of Markoff numbers below a given bound

Abstract

According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than 1 3 \frac {1}{3} of the square root of the discriminant) are in 1 : 1 correspondence with the solutions of the Diophantine equation p 2 + q 2 + r 2 = 3 p q r {p^2} + {q^2} + {r^2} = 3pqr . By relating Markoffs algorithm for finding solutions of this equation to a problem of counting lattice points in triangles, it is shown that the number of solutions less than x equals C log 2 3 x + O ( log ⁡ x log ⁡ log 2 x ) C{\log ^2}3x + O(\log x\log {\log ^2}x) with an explicitly computable constant C = 0.18071704711507 … C = 0.18071704711507 \ldots Numerical data up to 10 1300 {10^{1300}} is presented which suggests that the true error term is considerably smaller.