Geodesic
Unisequences and nearest integer continued fraction midpoint criteria for Pell's equation
Journal of integer sequences, Tome 12 (2009) no. 6
Zbl   EuDML
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 $x^{2} - Dy^{2} = \pm 1$ in terms of the NICF-H expansion of $\sqrt D$ 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 $\sqrt D$ to its NICF-P expansion.
Classification : 11A55, 11Y65
Keywords: nearest integer continued fraction, period-length, midpoint criteria, unisequence 
Matthews,  Keith R. Unisequences and nearest integer continued fraction midpoint criteria for Pell's equation. Journal of integer sequences, Tome 12 (2009) no. 6. http://geodesic.mathdoc.fr/item/JIS_2009__12_6_a7/
@article{JIS_2009__12_6_a7,
     author = {Matthews,  Keith R.},
     title = {Unisequences and nearest integer continued fraction midpoint criteria for {Pell's} equation},
     journal = {Journal of integer sequences},
     year = {2009},
     volume = {12},
     number = {6},
     zbl = {1201.11011},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JIS_2009__12_6_a7/}
}
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