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On the number of Markoff numbers below a given bound

1982
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Mathematics of Computation
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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

doi:10.1090/s0025-5718-1982-0669663-7
fatcat:zuqu7p2yvvcxjni4znigysebiy