A305313 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.

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, 34, 2, 1, 13, 233, 169, 1, 5, 34, 2, 29, 1, 5, 194, 13, 89, 610, 29, 1, 194, 2, 169, 433, 1, 5, 13, 34, 89, 985

Offset: 1

Author

Wolfdieter Lang, Jun 25 2018

Keywords

• nonn

Comments

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.

Examples

			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), ...
		

Crossrefs

Cf. A002559, A305312, A305314.

Formula

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.