A305312 Discriminant a(n) of the indefinite binary quadratic Markoff form m(n)*F_{m(n)}(x, y) with m(n) = A002559(n), for n >= 1.

5, 32, 221, 1517, 7565, 10400, 71285, 257045, 338720, 488597, 1687397, 3348896, 8732021, 15800621, 22953677, 75533477, 157326845, 296631725, 376282400, 514518485, 741527357, 1078334240, 1945074605, 7391012837, 10076746685, 12768548000, 16843627085, 24001135925, 34830756896, 50658755621, 83909288237, 164358078917, 342312755621, 347220276512, 781553243021, 1636268213885, 2244540316037, 2379883179965, 3756053306912, 7713367517021

Offset: 1

Author

Wolfdieter Lang, Jun 26 2018

Keywords

• nonn
• easy

Comments

Subsequence of A079896.

For the Markoff form f_{m(n)}(x, y) = m(n)*F_{m(n)}(x, y) of Cassels (pp. 31-39), see the comments on A305310. Some references are given in A002559, A305308 and A305310.

f_m(x, y) is an indefinite binary quadratic form because the discriminant is positive.

a(n) is also the discriminant D(n) = a(n) of the indefinite binary quadratic form determining the Markoff triple MT(n) = (x(n), y(n), m(n)) if the largest member is m(n) = A002559(n) and x(n) <= y(n) <= m(n). This is the form x^2 - 3*m*x*y + y^2 = -m^2 (with dropped argument n), or in reduced version X^2 + b*X*Y - b*Y^2 = -m^2, with b = b(n) = 3*m(n) - 2, where X = X(n) = y(n) - x(n) and Y = Y(n) = y(n). The uniqueness of such Markoff triples MT(n) with given largest members m(n) is a conjecture.

To find reduced forms one needs f(n) := ceiling(sqrt(D(n))) which is 3*m(n) because (3*m-1)^2 < 9*m^2 - 4 < (3*m)^2, due to 6*m(n) > 5, for n >= 1.

If the forms for a Markoff triple with largest member m are numerated with n giving m as m(n) = A002559(n)as in the present entry then the uniqueness conjecture is assumed to be true. Otherwise certain m(n) will lead to several different forms. - Wolfdieter Lang, Jul 30 2018

Examples

			a(5) = 7565 because 9*29^2 - 4 = 7565.
		

References

• J. W. S. Cassels, An Introduction to Diophantine Approximation, Cambridge University Press, 1957, Chapter II, The Markoff Chain, pp. 18-44.

Crossrefs

Cf. A002559, A079896, A305308, A305310.

Formula

a(n) = 9*m(n)^2 - 4 = 9*A002559(n)^2 - 4, n >= 1.