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Abstract: It is known that if the period s(d) of the continued fraction expansion of # d satisfies s(d) 2, then all Newton's approximants dqn ) are convergents of # d, and moreover we have p2n+1 q2n+1 0. Motivated with this fact we define two numbers j = j(d, n) and b = b(d) by R n = p2n+1+2j q2n+1+2j if R n is a convergent of # d; b = 1 and R n is a convergent of # d}|. (Update)
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BibTeX entry: (Update)
A. Dujella, Newton's formula and continued fraction expansion of # d, Experiment. Math. 10 (2001), 125--131. http://citeseer.ist.psu.edu/dujella01newtons.html More
@misc{ dujella01newtons,
author = "A. Dujella",
title = "Newton's formula and continued fraction expansion of # d",
text = "A. Dujella, Newton's formula and continued fraction expansion of # d, Experiment.
Math. 10 (2001), 125--131.",
year = "2001",
url = "citeseer.ist.psu.edu/dujella01newtons.html" }
Citations (may not include all citations):42 Die Lehre von den Kettenbruchen (context) - Perron - 1954
15 Lecture Notes in Math (context) - Schmidt, Approximation - 1980
5 Elementary Theory of Numbers (context) - Sierpinski - 1987
4 Sur la methode d'approximation de Newton (context) - Mikusinski - 1954
4 A note on continued fractions of quadratic irrationals (context) - Elezovic - 1997
4 Continued fractions and Newton's approximants (context) - Komatsu - 1999
2 Continued fractions and series (context) - Clemens, Merrill et al. - 1995
2 A numerical inverstigation into the length of the period of .. (context) - Williams - 1981
2 Some periodic continued fractions with long periods (context) - Patterson, Williams - 1985
2 On continued fraction expansions for binomial quadratic surd.. (context) - Frank - 1962
2 The length of the period of the simple continued fraction of.. (context) - Cohn - 1977
2 On Newton's method of approximation (context) - Sharma - 1959
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