The Markoff conjecture states that given a positive integer , there is at most one triple of positive integers with that satisfies the equation . The conjecture is known to be true when 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 , every ambiguous form in the principal genus corresponds to a divisor of , then the conjecture is true. As a result, we obtain criteria in terms of the Legendre symbols of primes dividing under which the conjecture holds. We also state a conjecture for the quadratic field that is equivalent to the Markoff conjecture for .
La conjecture de Markoff dit qu’étant donné un entier positif il existe au plus un triplet d’entiers positifs tels que et satisfaisant l’équation . La conjecture est vraie pour 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 , toute forme ambige dans le genre principal correspond à un diviseur de alors la conjecture est vraie. Comme conséquence, nous obtenons un critère, en termes de symboles de Legendre des premiers divisant , pour lequel la conjecture est vraie. Nous énonçons également une conjecture pour le corps quadratique qui est équivalente à la conjecture de Markoff pour .
@article{JTNB_2009__21_3_757_0, author = {Srinivasan, Anitha}, title = {Markoff numbers and ambiguous classes}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {757--770}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {3}, year = {2009}, doi = {10.5802/jtnb.701}, zbl = {1209.11036}, mrnumber = {2605546}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.701/} }
TY - JOUR AU - Srinivasan, Anitha TI - Markoff numbers and ambiguous classes JO - Journal de théorie des nombres de Bordeaux PY - 2009 SP - 757 EP - 770 VL - 21 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.701/ DO - 10.5802/jtnb.701 LA - en ID - JTNB_2009__21_3_757_0 ER -
Srinivasan, Anitha. Markoff numbers and ambiguous classes. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, pp. 757-770. doi : 10.5802/jtnb.701. http://archive.numdam.org/articles/10.5802/jtnb.701/
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