Geodesic
Power Residue Criteria for Quadratic Units and the Negative Pell Equation
Canadian mathematical bulletin, Tome 46 (2003) no. 1, pp. 39-53

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DOI

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).
DOI : 10.4153/CMB-2003-004-1
Mots-clés : 11R11, 11R27 
Bülow, Tommy. Power Residue Criteria for Quadratic Units and the Negative Pell Equation. Canadian mathematical bulletin, Tome 46 (2003) no. 1, pp. 39-53. doi: 10.4153/CMB-2003-004-1
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     journal = {Canadian mathematical bulletin},
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