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)


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