Halfway to a solution of ${X}^{2}-D{Y}^{2}=-3$
Journal de théorie des nombres de Bordeaux, Volume 6 (1994) no. 2, pp. 421-457.

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 ${x}^{2}-D{y}^{2}=±1$. 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 ${x}^{2}-D{y}^{2}=±1$. 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.

Classification: 11A55
Keywords: continued fraction, ideal, quadratic form, ambiguous cycle
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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, Volume 6 (1994) no. 2, pp. 421-457. http://www.numdam.org/item/JTNB_1994__6_2_421_0/

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