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Power Residue Criteria for Quadratic Units and the Negative Pell Equation

Published online by Cambridge University Press:  20 November 2018

Tommy Bülow
Tommy Bülow*
Affiliation:
Department of Mathematics, University of Copenhagen, DK-2100 Copenhagen, Denmark, email: tommy@math.ku.dk
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Abstract

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Let $d\,>\,1$ be a square-free integer. Power residue criteria for the fundamental unit ${{\varepsilon }_{d}}$ of the real quadratic fields $\mathbb{Q}(\sqrt{d})$ modulo a prime $p$ (for certain $d$ and $p$) are proved by means of class field theory. These results will then be interpreted as criteria for the solvability of the negative Pell equation ${{x}^{2}}\,-\,d{{p}^{2}}{{y}^{2}}\,=\,-1$. The most important solvability criterion deals with all $d$ for which $\mathbb{Q}(\sqrt{-d})$ has an elementary abelian 2-class group and $p\,\equiv \,5$ (mod 8) or $p\,\equiv \,9$ (mod 16).

Keywords

11R1111R27
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 46 , Issue 1 , 01 March 2003 , pp. 39 - 53
Copyright
Copyright © Canadian Mathematical Society 2003

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