## On a problem of Eisenstein.(English)Zbl 0851.11058

Let $${\mathcal D}$$ denote the set of squarefree positive integers $$d \equiv 5 \bmod 8$$. Eisenstein’s fourth problem is to give a criterion to decide whether the equation $$x^2 - dy^2 = 4$$ is solvable in odd integers $$x,y$$. This question is related to the 3-divisibility of the class number of the order $$\mathbb{Z} [\sqrt d]$$. The Eisenstein set $${\mathcal E}$$ is defined to be the set of all $$d \in {\mathcal D}$$ for which no odd solution $$x,y$$ exists. The author first gives an estimate for the upper density of the set $${\mathcal E}$$. He is not able to get a positive lower bound for the lower density, but he shows anyhow that $$\# \{d \in {\mathcal E} : d \leq x\} \gg x^{1/2}$$. The method is based on proving some interesting counting results about cubic number fields.
Reviewer: V.Ennola (Turku)

### MSC:

 11R11 Quadratic extensions 11R16 Cubic and quartic extensions 11R29 Class numbers, class groups, discriminants
Full Text: