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A305310 Numbers k(n) used for Cassels's Markoff forms MF(n) corresponding to the conjectured unique Markoff triples MT(n) with maximal entry m(n) = A002559(n), for n >= 1.

%I A305310 #12 Jul 31 2018 10:02:51
%S A305310 0,1,2,5,12,13,34,70,75,89,179,233,408,507,610,1120,1597,2378,2673,
%T A305310 2923,3468,4181,6089,10946,13860,15571,16725,19760,23763,28657,39916,
%U A305310 51709,80782,75025,113922,162867,206855,196418,249755,353702
%N A305310 Numbers k(n) used for Cassels's Markoff forms MF(n) corresponding to the conjectured unique Markoff triples MT(n) with maximal entry m(n) = A002559(n), for n >= 1.
%C A305310 For these Markoff forms see Cassels, p. 31. A link to the two original Markoff references is given in A305308.
%C A305310 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.
%C A305310 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]
%C A305310 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, ...].
%C A305310 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.
%C A305310 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.
%C A305310 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.
%D A305310 Martin Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013.
%D A305310 J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.
%D A305310 Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 172-180 and  222-224.
%D A305310 Oskar Perron, Über die Approximation irrationaler Zahlen durch rationale, Sitzungsber. Heidelberger Akademie der Wiss., 1921, 4. Abhandlung, pp. 1-17 , and part II., 8. Abhandlung, pp.1-12, Carl Winters Universitätsbuchhandlung.
%H A305310 Wolfdieter Lang, <a href="/A305310/a305310.pdf">A Note on Markoff Forms Determining Quadratic Irrationals with Purely Periodic Continued Fractions</a>
%F A305310 a(n) = k(n) has been defined in terms of the (conjectured unique) ordered Markoff triple MT(n) = (m_1(n), m_2(n), m(n)) with m(n) = A002559(n) in the comment above as 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).
%e A305310 n = 5: a(5) = k(5) = 12 because m(5) = A002559(5) = 29 with the triple MT(5) = (2, 5, 29). Whence 2*k1(5) - 5 == 0 (mod 29) for k1(5) = 17 < 29, and 5*k2(5) - 2 == 0 (mod 29) leads to k2(5) = 12. The smaller value is k2(5) = k(5) = 12. This leads to the form coefficients MF(5) = [29, 63, -31].
%e A305310 The forms MF(n) = [m(n), 3*m(n) - k(n), l(n) - 3*k(n)] with l(n) := (k(n)^2 + 1)/m(n) begin: [1, 3, 1], [2, 4, -2], [5, 11, -5], [13, 29, -13], [29, 63, -31], [34, 76, -34], [89, 199, -89], [169, 367, -181], [194, 432, -196], [233, 521, -233], [433, 941, -463], [610, 1364, -610], [985, 2139, -1055], [1325, 2961, -1327], [1597, 3571, -1597], [2897, 6451, -2927], [4181, 9349, -4181], [5741, 12467, -6149], [6466, 14052, -6914], [7561, 16837, -7639] ... .
%e A305310 The quadratic irrationals xi(n) = ((2*k(n) - 3*m(n)) + sqrt(D(n)))/(2*m(n)) begin: (-3 + sqrt(5))/2, -1 + sqrt(2), (-11 + sqrt(221))/10, (-29 + sqrt(1517))/26, (-63 + sqrt(7565))/58, (-19 + 5*sqrt(26))/17, (-199 + sqrt(71285))/178, (-367 + sqrt(257045))/338, (-108 + sqrt(21170))/97, (-521 + sqrt(488597))/466, (-941 + sqrt(1687397))/866, (-341 + sqrt(209306))/305, (-2139 + sqrt(8732021))/1970, (-2961 + sqrt(15800621))/2650, (-3571 + sqrt(22953677))/3194, (-6451 + sqrt(75533477))/5794, (-9349 + sqrt(157326845))/8362, (-12467 + 5*sqrt(11865269))/11482, (-3513 + 5*sqrt(940706))/3233, (-16837 + sqrt(514518485))/15122, ... .
%e A305310 The invariant M(xi(n)) = Lagrange(n) numbers begin with n >=1: sqrt(5), 2*sqrt(2), (1/5)*sqrt(221), (1/13)*sqrt(1517), (1/29)*sqrt(7565), (10/17)*sqrt(26), (1/89)*sqrt(71285), (1/169)*sqrt(257045), (2/97)*sqrt(21170), (1/233)*sqrt(488597), (1/433)*sqrt(1687397), (2/305)*sqrt(209306), (1/985)*sqrt(8732021), (1/1325)*sqrt(15800621), (1/1597)*sqrt(22953677), (1/2897)*sqrt(75533477), (1/4181)*sqrt(157326845), (5/5741)*sqrt(11865269), (10/3233)*sqrt(940706), (1/7561)*sqrt(514518485), ... .
%Y A305310 Cf. A002559, A305308, A305311, A305312, A305313, A305314.
%K A305310 nonn
%O A305310 1,3
%A A305310 _Wolfdieter Lang_, Jun 26 2018