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

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

Halfway to a solution of

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

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

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

Abstract (VO)
Résumé

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.

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.

MR Zbl

Classification : 11A55
Keywords: continued fraction, ideal, quadratic form, ambiguous cycle

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     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},
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R. A. Mollin; A. J. Van der Poorten; H. C. Williams. 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. https://jtnb.centre-mersenne.org/item/JTNB_1994__6_2_421_0/

Bibliographie
Cité par

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