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Anitha Srinivasan
Markoff numbers and ambiguous classes
Journal de théorie des nombres de Bordeaux, 21 no. 3 (2009), p. 757-770
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Résumé - 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 $c$.
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