Abstract
Abstract
Using a recent breakthrough of Smith [18], we improve the results of Fouvry and Klüners [4, 5] on the solubility of the negative Pell equation. Let
$\mathcal {D}$
denote the set of positive squarefree integers having no prime factors congruent to
$3$
modulo
$4$
. Stevenhagen [19] conjectured that the density of d in
$\mathcal {D}$
such that the negative Pell equation
$x^2-dy^2=-1$
is solvable with
$x, y \in \mathbb {Z}$
is
$58.1\%$
, to the nearest tenth of a percent. By studying the distribution of the
$8$
-rank of narrow class groups
$\operatorname {\mathrm {Cl}}^+(d)$
of
$\mathbb {Q}(\sqrt {d})$
, we prove that the infimum of this density is at least
$53.8\%$
.
Publisher
Cambridge University Press (CUP)
Subject
Computational Mathematics,Discrete Mathematics and Combinatorics,Geometry and Topology,Mathematical Physics,Statistics and Probability,Algebra and Number Theory,Theoretical Computer Science,Analysis
Reference21 articles.
1. Redei reciprocity, governing fields and negative Pell.;Stevenhagen;Math. Proc. Camb. Phil. Soc.
2. On the number of prime factors of integers without large prime divisors
3. On the distribution of
$\mathsf{Cl}(K)\left[{l}^{\infty}\right]$
for degree
$l$
cyclic fields;Koymans;J. Eur. Math. Soc
4. The Distribution of Ω(n) among Numbers with No Large Prime Factors
5. On the 4-rank of class groups of quadratic number fields