# Markoff numbers and ambiguous classes

Anitha Srinivasan^{[1]}

- [1] Siddhartha College (Mumbai University) Mumbai, INDIA

Journal de Théorie des Nombres de Bordeaux (2009)

- Volume: 21, Issue: 3, page 757-770
- ISSN: 1246-7405

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topSrinivasan, Anitha. "Markoff numbers and ambiguous classes." Journal de Théorie des Nombres de Bordeaux 21.3 (2009): 757-770. <http://eudml.org/doc/10911>.

@article{Srinivasan2009,

abstract = {The Markoff conjecture states that given a positive integer $c$, there is at most one triple $(a, b, c)$ of positive integers with $a\le b\le c$ that satisfies the equation $a^2+b^2+c^2=3abc$. The conjecture is known to be true when $c$ is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant $d=9c^2-4$, every ambiguous form in the principal genus corresponds to a divisor of $3c-2$, then the conjecture is true. As a result, we obtain criteria in terms of the Legendre symbols of primes dividing $d$ under which the conjecture holds. We also state a conjecture for the quadratic field $\mathbb\{Q\}(\sqrt\{9c^2-4\})$ that is equivalent to the Markoff conjecture for $c$.},

affiliation = {Siddhartha College (Mumbai University) Mumbai, INDIA},

author = {Srinivasan, Anitha},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {Markoff number; Markoff conjecture},

language = {eng},

number = {3},

pages = {757-770},

publisher = {Université Bordeaux 1},

title = {Markoff numbers and ambiguous classes},

url = {http://eudml.org/doc/10911},

volume = {21},

year = {2009},

}

TY - JOUR

AU - Srinivasan, Anitha

TI - Markoff numbers and ambiguous classes

JO - Journal de Théorie des Nombres de Bordeaux

PY - 2009

PB - Université Bordeaux 1

VL - 21

IS - 3

SP - 757

EP - 770

AB - The Markoff conjecture states that given a positive integer $c$, there is at most one triple $(a, b, c)$ of positive integers with $a\le b\le c$ that satisfies the equation $a^2+b^2+c^2=3abc$. The conjecture is known to be true when $c$ is a prime power or two times a prime power. We present an elementary proof of this result. We also show that if in the class group of forms of discriminant $d=9c^2-4$, every ambiguous form in the principal genus corresponds to a divisor of $3c-2$, then the conjecture is true. As a result, we obtain criteria in terms of the Legendre symbols of primes dividing $d$ under which the conjecture holds. We also state a conjecture for the quadratic field $\mathbb{Q}(\sqrt{9c^2-4})$ that is equivalent to the Markoff conjecture for $c$.

LA - eng

KW - Markoff number; Markoff conjecture

UR - http://eudml.org/doc/10911

ER -

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