### Finding the class number h(d) of a real quadratic field

Here d > 1 is squarefree.

Here D is the field discriminant.

We locate all reduced irrationals of the form (b+√D)/(2|c|), where c is negative and 4c divides D-b^{2}. We use the PQa continued fraction algorithm of Lagrange to break the set into disjoint cycles, retaining one number from each cycle. Each reduced number then gives rise to a reduced form (a,b,c) of discriminant D, where a=(b^{2}-D)/(4c).

We are able to also determine if the Pell equation x^{2}-Dy^{2}=-4 has a solution, by using the fact that the equation is soluble iff at least one of the above cycles is odd.
(See note and Henri Cohen's *A course in computational number theory*, page 260, First Edition.)

*Last modified 13th May 2003*

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