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Let  D  be a cube-free integer and let      ε  0 {\varepsilon _0}       be the fundamental unit of the pure cubic field      Q ( D 3 ) \mathcal {Q}(\sqrt [3]{D})      . It is well known that Voronoi’s algorithm can be used to determine      ε  0 {\varepsilon _0}      . In this work several results concerning Voronoi’s algorithm in      Q ( D 3 ) \mathcal {Q}(\sqrt [3]{D})       are derived and it is shown how these results can be used to increase the speed of calculating      ε  0 {\varepsilon _0}       for many values of  D . Among these  D  values are those such that      D ( &gt; 3 ) D( &gt; 3)       is not a prime      ≡  8 ( mod 9 ) \equiv 8\; \pmod 9       and the class number of      Q ( D 3 ) \mathcal {Q}(\sqrt [3]{D})       is not divisible by 3. A frequency table of all class numbers not divisible by 3 for all      Q ( D 3 ) \mathcal {Q}(\sqrt [3]{D})       with      D &gt; 2 ×  10 5 D &gt; 2 \times {10^5}       is also presented.

Some results concerning Voronoĭ’s continued fraction over 𝑄(\root3\of{𝐷})