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

Abstract

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The Markoff conjecture states that given a positive integer c , there is at most one triple ( a , b , c ) of positive integers with a ≤ b ≤ c that satisfies the equation a 2 + b 2 + c 2 = 3 a b c . 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 = 9 c 2 - 4 , every ambiguous form in the principal genus corresponds to a divisor of 3 c - 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 ℚ ( 9 c 2 - 4 ) that is equivalent to the Markoff conjecture for c .

How to cite

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Srinivasan, 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 -

References

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