Applications of a continued fraction algorithm to some class number problems
Michael D. Hendy · Mathematics of Computation · 1974
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.