Markoff numbers and ambiguous classes

Srinivasan, Ashok

doi:10.5802/jtnb.701

Cited by 5 publications

(3 citation statements)

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“…Part 1 follows because the fundamental unit in this case has positive norm and part 2 follows from [8,Lemma 4.4].…”

Section: A Form F Represents An Integer N If and Only If F ∼ (N B Cmentioning

confidence: 96%

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An improvement of the Minkowski bound for real quadratic orders using the Markoff theorem

Srinivasan

2011

Journal of Number Theory

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Using the Markoff theorem on indefinite binary quadratic forms it is shown that every element of the class group of any real quadratic order of discriminant d has an ideal of norm less than or equal to 1 + √ d 3 . This bound is best possible as there are quadratic orders of discriminant d which have ideal classes where the least norm is 1 + √ d 3 . These discriminants are explicitly given and their connection to the Markoff conjecture is highlighted.

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“…Part 1 follows because the fundamental unit in this case has positive norm and part 2 follows from [8,Lemma 4.4].…”

Section: A Form F Represents An Integer N If and Only If F ∼ (N B Cmentioning

confidence: 96%

“…Indeed there are very interesting connections between ambiguous forms, sums of squares and Markoff numbers as described in [6,8].…”

mentioning

confidence: 99%

An improvement of the Minkowski bound for real quadratic orders using the Markoff theorem

Srinivasan

2011

Journal of Number Theory

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“…The Markoff equation In this way, we can obtain infinitely many solution of Markoff equation, which we also known as Markoff triples. In ascending order the first 10 Markoff triples are (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) and the first 10 Markoff numbers are 1, 2, 5,13,29,34,89,169,194,233. It is easy to see that if (x, y, z) is a Markoff triple with x y z and z > 2 then x < y < z.…”

Section: Introductionmentioning

confidence: 99%

On the Unicity Conjecture for Generalized Markoff Equation

Gaire¹

2018

J. Adv. Coll. Engin. Mgt.

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The Markoff equation x2 + y2 + z2 = 3xyz is introduced by A.A. Markoff in 1879. A famous conjecture on the Markoff equation, made by Frobinus in 1913, states that any Markoff triples (x, y, z) with x ≤ y ≤ z is uniquely determined by its largest number z. The complete solution of this equation is still open however the partial solution is given by Barager (1996), Button (2001), Zhang (2007), Srinivasan (2009), Chen and Chen (2013). In 1957, Mordell developed a generalization to the Markoff equation of the form x2 + y2 + z2 = Axyz + B where, A and B are positive integers. In 2015, Donald McGinn take a particular form of above equation with A = 1 and B = A and gave a partial solution to the unicity conjecture to this equation. In this paper, the partial solution to the unicity conjecture to the equation of the form x2 + y2 + z2 =3xyz + A where A is positive integer with A ≤ 4(x2 –1) is given. Journal of Advanced College of Engineering and Management, Vol. 3, 2017, Page : 137-145

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The First Years

Narkiewicz

2012

Springer Monographs in Mathematics

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Markoff numbers and ambiguous classes

Cited by 5 publications

References 11 publications

An improvement of the Minkowski bound for real quadratic orders using the Markoff theorem

An improvement of the Minkowski bound for real quadratic orders using the Markoff theorem

On the Unicity Conjecture for Generalized Markoff Equation

The First Years

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