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Mathematics of Computation

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Some results concerning Voronoĭ's continued fraction over $ {\bf Q}(\root 3\of{D})$

Author: H. C. Williams
Journal: Math. Comp. 36 (1981), 631-652
MSC: Primary 12A45
MathSciNet review: 606521
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Abstract: Let D be a cube-free integer and let $ {\varepsilon _0}$ be the fundamental unit of the pure cubic field $ \mathcal{Q}(\sqrt[3]{D})$. It is well known that Voronoi's algorithm can be used to determine $ {\varepsilon _0}$. In this work several results concerning Voronoi's algorithm in $ \mathcal{Q}(\sqrt[3]{D})$ are derived and it is shown how these results can be used to increase the speed of calculating $ {\varepsilon _0}$ for many values of D. Among these D values are those such that $ D( > 3)$ is not a prime $ \equiv 8\; \pmod 9$ and the class number of $ \mathcal{Q}(\sqrt[3]{D})$ is not divisible by 3. A frequency table of all class numbers not divisible by 3 for all $ \mathcal{Q}(\sqrt[3]{D})$ with $ D < 2 \times {10^5}$ is also presented.

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Article copyright: © Copyright 1981 American Mathematical Society