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**
Unisequences and Nearest Integer Continued Fraction
Midpoint Criteria for Pell's Equation
**

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Keith R. Matthews

Department of Mathematics

University of Queensland

Brisbane 4072

Australia

and

Centre for Mathematics and its Applications

Australian National University

Canberra ACT 0200

Australia

**Abstract:**

The nearest integer continued fractions of Hurwitz, Minnegerode
(NICF-H) and in Perron's book *Die Lehre von den Kettenbrüchen*
(NICF-P)
are closely related.
Midpoint criteria for solving Pell's equation
*x*^{2} - *Dy*^{2} = ± 1 in terms
of the NICF-H expansion of √*D*
were derived by H. C. Williams using
singular continued fractions. We derive these criteria without the
use of singular continued fractions. We use an algorithm for
converting the regular continued fraction expansion of √*D* to
its NICF-P expansion.

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Received May 18 2009;
revised version received September 3 2009.
Published in *Journal of Integer Sequences*, September 22 2009.

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