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

1,5

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.

Links

Table of n, a(n) for n=1..60.

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.

Example

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.

Sequence in context: A143364 A309522 A336878 * A159046 A029937 A368926

Adjacent sequences: A305310 A305311 A305312 * A305314 A305315 A305316

Keyword

nonn

Author

Wolfdieter Lang, Jun 25 2018

Status

approved