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

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.