Halfway to a solution of $X^2 - DY^2 = -3$

Mollin, R. A.; Van der Poorten, A. J.; Williams, H. C.

Halfway to a solution of

X^{2} - D Y^{2} = - 3

Mollin, R. A. ; Van der Poorten, A. J. ; Williams, H. C.

Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 421-457.

Résumé
Abstract

Il est bien connu que le développement en fraction continue de $\sqrt{D}$ donne facilement le milieu du cycle principal des idéaux, c’est à dire le point à mi-parcourt d’une solution de $x^{2} - D y^{2} = \pm 1$ . Nous montrons ici que de façon analogue le point à mi-parcourt d’une solution de $x^{2} - D y^{2} = - 3$ peut-être reconnu. Nous expliquons ce qu’il en est.

It is well known that the continued fraction expansion of $\sqrt{D}$ readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of $x^{2} - D y^{2} = \pm 1$ . Here we notice that, analogously, the point halfway to a solution of $x^{2} - D y^{2} = - 3$ can be recognised. We explain what is going on.

MR Zbl

Classification : 11A55
Mots-clés : continued fraction, ideal, quadratic form, ambiguous cycle

@article{JTNB_1994__6_2_421_0,
     author = {Mollin, R. A. and Van der Poorten, A. J. and Williams, H. C.},
     title = {Halfway to a solution of $X^2 - DY^2 = -3$},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {421--457},
     publisher = {Universit\'e Bordeaux I},
     volume = {6},
     number = {2},
     year = {1994},
     mrnumber = {1360654},
     zbl = {0820.11015},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1994__6_2_421_0/}
}

TY  - JOUR
AU  - Mollin, R. A.
AU  - Van der Poorten, A. J.
AU  - Williams, H. C.
TI  - Halfway to a solution of $X^2 - DY^2 = -3$
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1994
SP  - 421
EP  - 457
VL  - 6
IS  - 2
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_1994__6_2_421_0/
LA  - en
ID  - JTNB_1994__6_2_421_0
ER  -

%0 Journal Article
%A Mollin, R. A.
%A Van der Poorten, A. J.
%A Williams, H. C.
%T Halfway to a solution of $X^2 - DY^2 = -3$
%J Journal de théorie des nombres de Bordeaux
%D 1994
%P 421-457
%V 6
%N 2
%I Université Bordeaux I
%U http://www.numdam.org/item/JTNB_1994__6_2_421_0/
%G en
%F JTNB_1994__6_2_421_0

Mollin, R. A.; Van der Poorten, A. J.; Williams, H. C. Halfway to a solution of $X^2 - DY^2 = -3$. Journal de théorie des nombres de Bordeaux, Tome 6 (1994) no. 2, pp. 421-457. http://www.numdam.org/item/JTNB_1994__6_2_421_0/

Bibliographie
Cité par

[1] H.W. Lenstra Jr, On the calculation of regulators and class numbers of quadratic fields, J. V. ARMITAGE ed., Journées Arithmétiques 1980, LMS Lecture Notes 56, Cambridge, 1982, pp. 123-151.. | MR | Zbl

[2] R.A. Mollin and A.J. Van Der Poorten, A note on symmetry and ambiguity, Bull. Austral. Math. Soc. 51 (1995), 215-233. | MR | Zbl

[3] Oskar Perron, Die Lehre von den Kettenbrüchen, (Chelsea reprint of 1929 edition). | Zbl

[4] A.J. Van Der Poorten, An introduction to continued fractions, Diophantine Analysis, LMS Lecture Notes in Math. 109, ed. J. H. LOXTON and A. J. VAN DER POORTEN, Cambridge University Press, 1986, pp. 99-138. | MR | Zbl

[5] A.J. Van Der Poorten, Fractions of the period of the continued fraction expansion of quadratic integers, Bull. Austral. Math. Soc 44 (1991), 155-169. | MR | Zbl

[6] D. Shanks, Class number, a theory of factorization, and genera, Proc. Symp. Pure Math., 20 (1969 Institute on Number Theory), Amer. Math. Soc., Providence 1971, pp. 415-440, see also The infrastructure of a real quadratic field and its applications, Proc. Number Theory Conference, Boulder, 1972. | MR | Zbl

[7] D. Shanks, On Gauβ and composition, Number Theory and Applications, Richard A. MOLLIN ed. (NATO - Advanced Study Institute, Banff, 1988), Kluwer Academic Publishers, Dordrecht, 1989, pp. 163-204.