Journal of Integer Sequences, Vol. 12 (2009), Article 09.6.7

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

Keith R. Matthews
Department of Mathematics
University of Queensland
Brisbane 4072
Centre for Mathematics and its Applications
Australian National University
Canberra ACT 0200


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 x2 - Dy2 =   ± 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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