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 ω \omega , where ω = √ d \omega = \surd d for d ≡ 2 , 3 ( mod 4 ) d \equiv 2,3 \pmod 4 and ( √ d − 1 ) / 2 (\surd d - 1)/2 for d ≡ 1 ( mod 4 ) d \equiv 1 \pmod 4 . The recursively generated terms Q n {Q_n} in his algorithm lead to all norms of primitive algebraic integers of Q ( √ d ) Q(\surd d) less than √ ( D / 4 ) \surd (D/4) , D being the discriminant. By ensuring that the values Q n {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.