English

The negative Pell equation

Number Theory 2019-08-08 v2

Abstract

By applying methods recently developed by A. Smith with regards to Goldfeld's conjecture, we show that the density of square-free integers dd in [1,N][1, N] for which the negative Pell equation x2dy2=1x^2 - dy^2 = -1 has a solution is as predicted by the model of Stevenhagen.

Cite

@article{arxiv.1907.13105,
  title  = {The negative Pell equation},
  author = {Erick Knight and Stanley Yao Xiao},
  journal= {arXiv preprint arXiv:1907.13105},
  year   = {2019}
}

Comments

We made a mistake regarding applying A. Smith's methods, in particular eq. (9.8) in our paper is not valid

R2 v1 2026-06-23T10:35:11.627Z