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Halfway to a solution of X 2 - D Y 2 = - 3

R. A. Mollin, A. J. Van der Poorten, H. C. Williams (1994)

Journal de théorie des nombres de Bordeaux

It is well known that the continued fraction expansion of 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.

Ideal Criteria for both Ideal Criteria for both X2-dy2 = M1 And X2-dy2 = M2 to have Primitive Solutions for any Integers M1, M2 Prime to D > 0

Mollin, R. (2002)

Serdica Mathematical Journal

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].

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