Markoff numbers and ambiguous classes
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 3, p. 757-770

The Markoff conjecture states that given a positive integer c, there is at most one triple (a,b,c) of positive integers with abc 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 (9c 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 abc et satisfaisant l’équation a 2 +b 2 +c 2 =3abc. 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=9c 2 -4, toute forme ambige dans le genre principal correspond à un diviseur de 3c-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 (9c 2 -4) qui est équivalente à la conjecture de Markoff pour c.

@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},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {3},
     year = {2009},
     pages = {757-770},
     doi = {10.5802/jtnb.701},
     mrnumber = {2605546},
     zbl = {1209.11036},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_3_757_0}
}
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://www.numdam.org/item/JTNB_2009__21_3_757_0/

[1] A. Baragar, On the unicity conjecture for Markoff numbers. Canad. Math. Bull. 39 (1996), 3–9. | MR 1382484 | Zbl 0846.11020

[2] E. Bombieri, Continued fractions and the Markoff tree. Expo. Math. 25 (2007), no. 3, 187–213 | MR 2345177 | Zbl 1153.11030

[3] J. O. Button, The uniqueness of prime Markoff numbers. Bull. London Math. Soc., 58 (1998), 9–17. | MR 1666058 | Zbl 0932.11020

[4] J. O. Button, Markoff numbers, principal ideals and continued fraction expansions. Journal of Number Theory, 87 (2001), 77–95. | MR 1816037 | Zbl 0983.11040

[5] J. W. S. Cassels, An introduction to Diophantine approximation. Cambridge University Press, 1957. | MR 87708 | Zbl 0077.04801

[6] H. Cohen, A course in computational algebraic number theory. Springer-Verlag, 1993. | MR 1228206 | Zbl 0786.11071

[7] H. Cohn, Advanced Number Theory. Dover Publications, 1980. | MR 594936 | Zbl 0474.12002

[8] M. L. Lang, S. P. Tan, A simple proof of the Markoff conjecture for prime powers. Geom. Dedicata 129 (2007), 15–22. | MR 2353978 | Zbl 1133.11023

[9] A. A. Markoff, Sur les formes quadratiques binaires indéfinies I. Math. Ann. 15 (1879), 381–409.

[10] R. A. Mollin, Quadratics. CRC press, Boca Raton, 1996. | MR 1383823 | Zbl 0858.11001

[11] S. Perrine, Sur une généralisation de la théorie de Markoff. Journal of Number Theory 37 (1991), 211–230. | MR 1092607 | Zbl 0714.11039

[12] S. Perrine, Un arbre de constantes d’approximation analogue à celui de l’équation diophantienne de Markoff. Journal de Théorie des Nombres de Bordeaux, 10, no. 2, (1998), 321–353. | Numdam | MR 1828249 | Zbl 0924.11057

[13] P. Ribenboim, My Numbers, My Friends, Popular Lectures on Number Theory. Springer-Verlag, 2000. | MR 1761897 | Zbl 0947.11001

[14] P. Schmutz, Systoles of arithmetic surfaces and the Markoff spectrum. Math. Ann. 305 (1996), no. 1, 191–203. | MR 1386112 | Zbl 0853.11054

[15] A. Srinivasan, A note on the Markoff conjecture. Biblioteca de la Revista Matemática Iberoamericana, Proceedings of the “Segundas Jornadas de Teoría de Números” (Madrid, 2007), pp. 253–260. | Zbl pre05666910

[16] D. Zagier, On the number of Markoff numbers below a given bound. Math. Comp. 39 (1982), 709–723 | MR 669663 | Zbl 0501.10015

[17] Y. Zhang, Conguence and uniqueness of certain Markoff numbers. Acta Arith. 128 (2007), no. 3, 295–301. | MR 2313995 | Zbl 1144.11030

[18] Y. Zhang, An elementary proof of uniqueness of Markoff numbers which are prime powers. Preprint, arXiv:math.NT/0606283 (version 2).