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For these Markoff forms see Cassels, p. 31. A link to the two original Markoff references is given in A305308.

MF(n) = f_{m(n)}(x, y) = m(n)*F_{m(n)}(x, y) = m(n)*x^2 + (3*m(n) - 2*k(n))*x*y + (l(n) - 3*k(n))*y^2, with the Markoff number m = m(n) = A002559(n) and l(n) = (k(n)^2 + 1)/m(n), for n >= 1.

Every m(n) is proved to appear as largest member of a Markoff triple MT(n) = (m_1(n), m_2(n), m(n)), with positive integers m_1(n) < m_2(n) < m(n) for n >= 3 (MT(1) = (1, 1, 1) and MT(2) = (1, 1, 2)) satisfying the Markoff equation m_1(n)^2 + m_2(n)^2 + m(n)^2 = 3*m_1(n)*m_2(n)*m(n). The famous Markoff uniqueness conjecture is that m(n) as largest member determines exactly one ordered triple MT(n). See, e.g., the Aigner reference, pp. 38-39, and Corollary 3.5, p. 48. [In numerating the sequence with n related to A002559(n) this conjecture is assumed to be true. - Wolfdieter Lang, Jul 29 2018]

The nonnegative integers k(n) are defined for the Markoff forms given by Cassels by k(n) = min{k1(n), k2(n)}, where m_1(n)*k1(n) - m_2(n) == 0 (mod m(n)), with 0 <= k1(n) < m(n), and m_2(n)*k2(n) - m_1(n) == 0 (mod m(n)), with 0 <= k2(n) < m(n). The k1 and k2 sequences are k1 = [0, 1, 2, 5, 17, 13, 34, 99, 119, 89, 179, 233, 577, 818, 610, 1777, 1597, 3363, 2673, 2923, 5609, 4181, 6089, 10946, 19601, 22095, 26536, 31881, 38447, 28657, 39916, 51709, 114243, 75025, 113922, 263522, 206855, 196418, 396263, 572063, ...], and k2 = [0, 1, 3, 8, 12, 21, 55, 70, 75, 144, 254, 377, 408, 507, 987, 1120, 2584, 2378, 3793, 4638, 3468, 6765, 8612, 17711, 13860, 15571, 16725, 19760, 23763, 46368, 56641, 83428, 80782, 121393, 180763, 162867, 292538, 317811, 249755, 353702, ...].

The discriminant of the form MF(n) = f_{m(n)}(x, y) is D(n) = 9*m(n)^2 - 4. D(n) = A305312(n), for n >= 1. Because D(n) > 0 (not a square) this is an indefinite binary quadratic form, for n >= 1. See Cassels Fig. 2 on p. 32 for the Markoff tree with these forms.

The quadratic irrational xi, determined by the solution with positive square root of f_{m(n)}(x, 1) = 0, is xi(n) = ((2*k - 3*m) + sqrt(D))/(2*m) (the argument n has been dropped). The regular continued fraction is eventually periodic, but not purely periodic. One can find equivalent Markoff forms determining purely periodic quadratic irrationals. The corresponding k sequence is given in A305311.

For the approximation of xi(n) with infinitely many rationals (in lowest terms) Perron's unimodular invariant M(xi) enters. For quadratic irrationals M(xi) < 3, and the values coincide with the discrete Lagrange spectrum < 3: M(xi(n)) = Lagrange(n) = sqrt{D(n)}/m(n), n >= 1. For n=1..4 see A002163, A010466, A200991 and A305308.

For every irrational number alpha not equivalent to each of the following three numbers i) golden section A001622, ii) sqrt(2) = A002193 and iii) (5 + sqrt(221))/14 = A177841 there exist infinitely many rational numbers h/k (in lowest terms) such that |alpha - h/k| < 1/(Lagrange(4)*k^2). The constant L(4) cannot be replaced by a larger number because then the statement becomes false for, e.g., alpha = (23 + sqrt(1517))/26. Two real numbers x and y are equivalent if there exist integers p, q, r and s with |p*s - q*r| = 1 such that y = (p*x + q)/(r*x + s) (unimodular transformation). This means that the continued fractions of x and y become eventuakky identical.

See the references (in Havil, p. 174, equivalence classes of numbers should have been excluded).

The continued fraction of Lagrange(4) is [2; repeat(1, 252, 3, 1012, 3, 252, 1, 4)]. 1/L(4) = 0.3337725078... < 1/3.

Perron's numbers M(xi) (pp. 4, 5), for M(xi) < 3, are the Lagrange numbers sqrt(9*Q^2 - 4)/Q, with Q = Q(n) = A002559(n), n >= 1, and his corresponding xi(4) = (sqrt(1517) + 23)/26 with a purely periodic simple continued fraction [repeat(2, 2, 1, 1, 1, 1)].

Cassels (p. 18) uses the version: For irrational theta not equivalent to the above given three numbers i), ii) and iii) there are infinitely many solutions of q*||q*theta|| < 1/Lagrange(4), where 1/Lagrange(4) cannot be improved for theta equivalent to -29/26 + (1/26)*sqrt(1517). Here ||x|| is the positive difference between x and the nearest integer.