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Mathematics of Computation

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Applications of a continued fraction algorithm to some class number problems

Author: M. D. Hendy
Journal: Math. Comp. 28 (1974), 267-277
MSC: Primary 12A50; Secondary 10F20, 12A25
MathSciNet review: 0330102
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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.

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Keywords: Principal ideals, real quadratic field, fundamental unit, infinite continued fraction, Lagrange algorithm, class number, genera, Shanks sequence $ {S_n}$
Article copyright: © Copyright 1974 American Mathematical Society

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