On the negative Pell equation

Étienne Fouvry; Jürgen Klüners

doi:http://doi.org/10.4007/annals.2010.172.2035

Abstract

We give asymptotic upper and lower bounds for the number of squarefree $d$ ($0\lt d\leq X$) such that the equation $x^2-dy^2=-1$ is solvable. These estimates, as usual, can equivalently be interpreted in terms of real quadratic fields with a fundamental unit with norm $-1$ and give strong evidence in the direction of a conjecture due to P. Stevenhagen.

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@article {Sh, MRKEY = {552470}, AUTHOR = {Shiu, P.}, TITLE = {A {B}run-{T}itchmarsh theorem for multiplicative functions}, JOURNAL = {J. Reine Angew. Math.}, FJOURNAL = {Journal für die Reine und Angewandte Mathematik}, VOLUME = {313}, YEAR = {1980}, PAGES = {161--170}, ISSN = {0075-4102}, CODEN = {JRMAA8}, MRCLASS = {10H25}, MRNUMBER = {81h:10065}, MRREVIEWER = {A. I. Vinogradov}, DOI = {10.1515/crll.1980.313.161}, ZBLNUMBER = {0412.10030}, }
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@article {St, MRKEY = {1259426}, AUTHOR = {Stevenhagen, Peter}, TITLE = {The number of real quadratic fields having units of negative norm}, JOURNAL = {Experiment. Math.}, FJOURNAL = {Experimental Mathematics}, VOLUME = {2}, YEAR = {1993}, NUMBER = {2}, PAGES = {121--136}, ISSN = {1058-6458}, MRCLASS = {11R11 (11R27 11Y40)}, MRNUMBER = {94k:11120}, MRREVIEWER = {Duncan A. Buell}, URL = {http://projecteuclid.org/getRecord?id=euclid.em/1048516217}, ZBLNUMBER = {0792.11041}, }
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@book {We, MRKEY = {734177}, AUTHOR = {Weil, Andr{é}}, TITLE = {Number Theory{\rm :} An Approach Through History, From Hammurapi to Legendre}, PUBLISHER = {Birkhäuser}, ADDRESS = {Boston, MA}, YEAR = {1984}, PAGES = {xxi+375}, ISBN = {0-8176-3141-0}, MRCLASS = {01A05 (11-03)}, MRNUMBER = {85c:01004}, MRREVIEWER = {Ezra Brown}, DOI = {10.1007/978-0-8176-4571-7}, ZBLNUMBER = {0531.10001}, }

On the negative Pell equation

Abstract

Authors