Applications of a continued fraction algorithm to some class number problems

Abstract:

We make extensive use of Lagrange’s algorithm for the evaluation of the quotients in the continued fraction expansion of the quadratic surd $\omega$, where $\omega = \surd d$ for $d \equiv 2,3 \pmod 4$ and $(\surd d - 1)/2$ for $d \equiv 1 \pmod 4$. The recursively generated terms ${Q_n}$ in his algorithm lead to all norms of primitive algebraic integers of $Q(\surd d)$ less than $\surd (D/4)$, D being the discriminant. By ensuring that the values ${Q_n}$ contain at most one small prime, we are able to generate sequences of determinants d of real quadratic fields whose genera usually contain more than one ideal class. Formulae for their fundamental units are given.