On the number of Markoff numbers below a given bound

Don Zagier
1982 Mathematics of Computation  
According to a famous theorem of Markoff, the indefinite quadratic forms with exceptionally large minima (greater than f of the square root of the discriminant) are in 1 : 1 correspondence with the solutions of the Diophantine equation p2 + q2 + r1 = ~ipqr. 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 Clog2 3x + 0(log x log log2 x) with an explicitly
more » ... explicitly computable constant C = 0.18071704711507.... Numerical data up to 101300 is presented which suggests that the true error term is considerably smaller.
doi:10.1090/s0025-5718-1982-0669663-7 fatcat:zuqu7p2yvvcxjni4znigysebiy