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%I #9 Jun 28 2018 13:06:18
%S 1,1,1,1,2,1,1,2,5,1,5,1,2,13,1,5,1,2,5,13,34,1,29,1,2,29,5,13,89,1,5,
%T 34,2,1,13,233,169,1,5,34,2,29,1,5,194,13,89,610,29,1,194,2,169,433,1,
%U 5,13,34,89,985
%N Smallest member m_1(n) of the ordered Markoff triple MT(n) with largest member m(n) = A002559(n), n >= 1. These triples are conjectured to be unique.
%C The second member m_2 of the Markoff (Markov) triple MT(n) = (m_1(n), m_2(n), m(n)) with m_1(n) <= m_2(n) <= m(n), for n >= 1, with m(n) = A002559(n) is given in A305314(n). For n>=3 the inequalities are strict. The existence of MT(n) with largest number m(n) is proved. The uniqueness is conjectured. The Markoff equation is (the argument n is dropped) m_1^2 + m_2^2 + m^2 = 3*m_1*m_2*m. See the references under A002559.
%F a(n) = m_1(n) is the fundamental proper solution x of the indefinite binary quadratic form x^2 - 3*m(n)*x*y + y^2, of discriminant D(n) = 9*m(n)^2 - 4 = A305312(n), representing -m(n)^2, for n >= 1, with x <= y. The uniqueness conjecture means that there are no other such fundamental solutions.
%e The Markoff triples begin: (1, 1, 1), (1, 1, 2), (1, 2, 5), (1, 5, 13), (2, 5, 29), (1, 13, 34), (1, 34, 89), (2, 29, 169), (5, 13, 194), (1, 89, 233), (5, 29, 433), (1, 233, 610), (2, 169, 985), (13, 34, 1325), (1, 610, 1597), (5,194,2897), (1, 1597, 4181), (2, 985, 5741), (5, 433, 6466), (13, 194, 7561), (34, 89, 9077), ...
%Y Cf. A002559, A305312, A305314.
%K nonn
%O 1,5
%A _Wolfdieter Lang_, Jun 25 2018
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