On the Number of Markoff Numbers Below a Given Bound

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 computable constant C = 0.18071704711507.... Numerical data up to 101300 is presented which suggests that the true error term is considerably smaller. 1. By a Markoff triple we mean a solution (p, q, r) of the Markoff equation (1) p2 + q2 + r2 = 3pqr (p, q, r E Z, 1 < p < q < r); a Markoff number is a member of such a triple. These numbers, of which the first few are 1, 2, 5, 13, 29, 34, 89, 169, 194, 233, 433, 610, 985,..., play a role in a famous theorem of Markoff [10] (see also Frobenius [6], Cassels [2]): the GL2(Z)-equivalence classes of real indefinite binary quadratic forms Q of discriminant 1 for which the invariant p(Q)= min \Q{x,y)\ (i,j)ez2-{(0,0)} is greater than ^ are in one-to-one correspondence with the Markoff triples, the invariant p(Q) for the form corresponding to (p, q, r) being (9 — 4r'2)'x/2. Thus the part of the Markoff spectrum (the set of all p(Q)) lying above y is described exactly by the Markoff numbers. An equivalent theorem is that, under the action of SL2(Z) on R U {oo} given by x -> (ax + b)/(cx + d), the SL2(Z)-equivalence classes of real numbers x for which the approximation measure p(x) = lim sup ( q ■ min I qx — p I is > y are in 1 : 1 correspondence with the Markoff triples, the spectrum being the same as above (e.g. p(x) = 5~'/2 for x equivalent to the golden ratio and p(x) < 8"1/2 for all other x). Thus the Markoff numbers are important both in the theory of quadratic forms and in the theory of Diophantine approximation. They have also arisen in connection with problems in several other branches of mathematics, e.g. the Received July 15, 1981; revised December 16, 1981. 1980 Mathematics Subject Classification. Primary 10B10; Secondary 10A21, 10C25, 10J25, 10F99. 709 ©1982 American Mathematical Society 0025-5718/82/0OOO-O259/SO3.75 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use