Markoff numbers and ambiguous classes

Anitha Srinivasan

doi:10.5802/jtnb.701

Anitha Srinivasan ¹

¹ Siddhartha College (Mumbai University) Mumbai, INDIA

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 757-770.

Abstract (VO)
Résumé

The Markoff conjecture states that given a positive integer $c$ , there is at most one triple $(a, b, c)$ of positive integers with $a \leq b \leq c$ 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 $ℚ (\sqrt{9 c^{2} - 4})$ that is equivalent to the Markoff conjecture for $c$ .

La conjecture de Markoff dit qu’étant donné un entier positif $c$ il existe au plus un triplet $(a, b, c)$ d’entiers positifs tels que $a \leq b \leq c$ et satisfaisant l’équation $a^{2} + b^{2} + c^{2} = 3 a b c$ . La conjecture est vraie pour $c$ une puissance d’un nombre premier ou deux fois une puissance d’un nombre premier. Nous présentons une preuve élémentaire de ce résultat. Nous montrons également que si, dans le groupe des classes des formes de discriminant $d = 9 c^{2} - 4$ , toute forme ambige dans le genre principal correspond à un diviseur de $3 c - 2$ alors la conjecture est vraie. Comme conséquence, nous obtenons un critère, en termes de symboles de Legendre des premiers divisant $d$ , pour lequel la conjecture est vraie. Nous énonçons également une conjecture pour le corps quadratique $ℚ (\sqrt{9 c^{2} - 4})$ qui est équivalente à la conjecture de Markoff pour $c$ .

MR Zbl

DOI : 10.5802/jtnb.701

Affiliations des auteurs :

Anitha Srinivasan ¹

¹ Siddhartha College (Mumbai University) Mumbai, INDIA

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Anitha Srinivasan. Markoff numbers and ambiguous classes. Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 3, pp. 757-770. doi : 10.5802/jtnb.701. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.701/

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Władysław Narkiewicz The First Years, Rational Number Theory in the 20th Century (2012), p. 13 | DOI:10.1007/978-0-85729-532-3_2
Anitha Srinivasan An improvement of the Minkowski bound for real quadratic orders using the Markoff theorem, Journal of Number Theory, Volume 131 (2011) no. 8, pp. 1420-1428 | DOI:10.1016/j.jnt.2011.02.008 | Zbl:1229.11147

Cité par 2 documents. Sources : Crossref, zbMATH