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.
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
KEYWORD
nonn
AUTHOR
Wolfdieter Lang, Jun 25 2018
STATUS
approved