Frobenius distributions for real quadratic orders. (English) Zbl 0847.11010

Stevenhagen, Peter

J. Théor. Nombres Bordx. 7, No. 1, 121-132 (1995).

In 1932, T. Nagell asked questions pertaining to the solutions of the Pell equation \((*)\) \(x^2- Dy^2= - 1\) for square-free \(D> 1\). Solutions to \((*)\) imply that \(D\) is a sum of two relatively prime squares. If \(\mathcal S\) denotes the set of all integers which are sums of two relatively prime squares, and \({\mathcal S}^-\) is the set of all integers \(D\) for which \((*)\) has integer solutions, then Nagell’s query boils down to: Does \({\mathcal S}^-\) have natural density in \(\mathcal S\)? This and an equivalent query of Rédei are answered by the author’s conjecture (too technical to state here). The author cites data, for which the conjecture holds in special cases, at the end of the paper. The general case remains open. The author also discusses possible generalizations from the quadratic case to abelian extensions of higher degree.

Reviewer: R.Mollin (Calgary)

MSC:

11D09	Quadratic and bilinear Diophantine equations
11R11	Quadratic extensions
11R45	Density theorems
11R21	Other number fields

Keywords:

solutions of the Pell equation; integers which are sums of two relatively prime squares; abelian extensions of higher degree

References:

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