Unisequences and Nearest Integer Continued Fraction
Midpoint Criteria for Pell's Equation
Keith R. Matthews
Department of Mathematics
University of Queensland
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
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.
Full version: pdf,
Received May 18 2009;
revised version received September 3 2009.
Published in Journal of Integer Sequences, September 22 2009.
Journal of Integer Sequences home page