The negative Pell equation

TL;DR

This paper demonstrates that the distribution of square-free integers for which the negative Pell equation has solutions aligns with Stevenhagen's predicted model, using advanced methods related to Goldfeld's conjecture.

Contribution

It applies recent methods by A. Smith to confirm Stevenhagen's model for the density of solutions to the negative Pell equation.

Findings

Density matches Stevenhagen's prediction

Methods confirm conjectural distribution

Supports Goldfeld's conjecture in this context

Abstract

By applying methods recently developed by A. Smith with regards to Goldfeld's conjecture, we show that the density of square-free integers $d$ in $[1, N]$ for which the negative Pell equation $x^2 - dy^2 = -1$ has a solution is as predicted by the model of Stevenhagen.