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A305310
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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.
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5
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0, 1, 2, 5, 12, 13, 34, 70, 75, 89, 179, 233, 408, 507, 610, 1120, 1597, 2378, 2673, 2923, 3468, 4181, 6089, 10946, 13860, 15571, 16725, 19760, 23763, 28657, 39916, 51709, 80782, 75025, 113922, 162867, 206855, 196418, 249755, 353702
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OFFSET
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1,3
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COMMENTS
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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.
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REFERENCES
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Martin Aigner, Markov's Theorem and 100 Years of the Uniqueness Conjecture, Springer, 2013.
J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.
Julian Havil, The Irrationals, Princeton University Press, Princeton and Oxford, 2012, pp. 172-180 and 222-224.
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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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].
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] ... .
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, ... .
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), ... .
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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