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Necessary and Sufficient Conditions for the Central Norm to Equal $2^h$ in the Simple Continued Fraction Expansion of $\sqrt{2^hc}$ for Any Odd $c>1$

  Published:2005-03-01
 Printed: Mar 2005
  • R. A. Mollin
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Abstract

We look at the simple continued fraction expansion of $\sqrt{D}$ for any $D=2^hc $ where $c>1$ is odd with a goal of determining necessary and sufficient conditions for the central norm (as determined by the infrastructure of the underlying real quadratic order therein) to be $2^h$. At the end of the paper, we also address the case where $D=c$ is odd and the central norm of $\sqrt{D}$ is equal to $2$.
Keywords: quadratic Diophantine equations, simple continued fractions, norms of ideals, infrastructure of real quadratic fields quadratic Diophantine equations, simple continued fractions, norms of ideals, infrastructure of real quadratic fields
MSC Classifications: 11A55, 11D09, 11R11 show english descriptions Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
Quadratic and bilinear equations
Quadratic extensions
11A55 - Continued fractions {For approximation results, see 11J70} [See also 11K50, 30B70, 40A15]
11D09 - Quadratic and bilinear equations
11R11 - Quadratic extensions
 

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