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# Criteria for Simultaneous Solutions of $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$

Published:2002-09-01
Printed: Sep 2002
• R. A. Mollin
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## Abstract

The purpose of this article is to provide criteria for the simultaneous solvability of the Diophantine equations $X^2 - DY^2 = c$ and $x^2 - Dy^2 = -c$ when $c \in \mathbb{Z}$, and $D \in \mathbb{N}$ is not a perfect square. This continues work in \cite{me}--\cite{alfnme}.
 Keywords: continued fractions, Diophantine equations, fundamental units, simultaneous solutions
 MSC Classifications: 11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15] 11R11 - Quadratic extensions 11D09 - Quadratic and bilinear equations