Groupes d'unités elliptiques
It is well known that the continued fraction expansion of readily displays the midpoint of the principal cycle of ideals, that is, the point halfway to a solution of . Here we notice that, analogously, the point halfway to a solution of can be recognised. We explain what is going on.
This article provides necessary and sufficient conditions for both of the Diophantine equations X^2 − DY^2 = m1 and x^2 − Dy^2 = m2 to have primitive solutions when m1 , m2 ∈ Z, and D ∈ N is not a perfect square. This is given in terms of the ideal theory of the underlying real quadratic order Z[√D].