Improving the Speed of Calculating the Regulator of Certain Pure Cubic Fields

To calculate R, the regulator of a pure cubic field $Q(\sqrt [3]{{D)}}$, a complete period of Voronoi’s continued fraction algorithm over $Q(\sqrt [3]{{D)}}$ is usually generated. In this paper it is shown how, in certain pure cubic fields, R can be determined by generating only about one third of this period. These results were used on a computer to find R and then the class number for all pure cubic fields $Q(\sqrt [3]{{p)}}$, where p is a prime, $p \equiv - 1\;\pmod 3$, and $p < 2 \times {10^5}$. Graphs illustrating the distribution of such cubic fields with class number one are presented.