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Mathematics of Computation

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Density computations for real quadratic units

Authors: Wieb Bosma and Peter Stevenhagen
Journal: Math. Comp. 65 (1996), 1327-1337
MSC (1991): Primary 11R11, 11Y40, 11R45; Secondary 11E16
MathSciNet review: 1344607
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Abstract: In order to study the density of the set of positive integers $d$ for which the negative Pell equation $x^{2}-dy^{2}=-1$ is solvable in integers, we compute the norm of the fundamental unit in certain well-chosen families of real quadratic orders. A fast algorithm that computes 2-class groups rather than units is used. It is random polynomial-time in $\log d$ as the factorization of $d$ is a natural part of the input for the values of $d$ we encounter. The data obtained provide convincing numerical evidence for the density heuristics for the negative Pell equation proposed by the second author. In particular, an irrational proportion $P=1-\prod _{j\ge 1 \text {\rm \ odd}} (1-2^{-j}) \approx .58$ of the real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 should have a fundamental unit of norm $-1$.

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Additional Information

Wieb Bosma
Affiliation: School of Mathematics and Statistics, University of Sydney, Sydney NSW 2006, Australia

Peter Stevenhagen
Affiliation: Faculteit Wiskunde en Informatica, Universiteit van Amsterdam, Plantage Muidergracht 24, 1018 TV Amsterdam, The Netherlands

Keywords: Real quadratic class groups, negative Pell equation, density theorems
Received by editor(s): February 24, 1995
Article copyright: © Copyright 1996 American Mathematical Society