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Density computations for real quadratic units. (English) Zbl 0859.11064

The authors perform a series of direct computations to verify the density of those nonsquare integers \(d(>0)\) for which the form over \(\mathbb{Z}\), \(Q= x^2-dy^2\), takes the (non-Pellian) value \(-1\). There are necessary conditions (modulo 4) and Rédei conditions on the 4-rank \((e)\) of the class group, which come so close that the questionable cases are few and far between [see the second author, CANT, Math. Appl., Dordr. 325, 187-200 (1995; Zbl 0838.11066)]. In case the value \(Q=-1\) is taken, then \(-Q\) and \(Q\) are equivalent; this is easier to verify from class structure than a computation of the (odd) period of a continued fraction of a unit. Since the conjectured density depends on the value of \(e\), the testing thins out very rapidly (as for even the small values of \(e\) the testing requires 50 Sun Workstations). Ultimately, earlier estimates of the density of cases where \(Q=-1\) (within those valid modulo 4) still remain \(\approx .57339\).
Reviewer: H.Cohn (Bowie)

MSC:

11Y40 Algebraic number theory computations
11R45 Density theorems
11R11 Quadratic extensions
11E16 General binary quadratic forms

Citations:

Zbl 0838.11066
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References:

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