Get 20M+ Full-Text Papers For Less Than $1.50/day. Start a 14-Day Trial for You or Your Team.

Learn More →

On the Lengths of the Periods of the Continued Fractions of Square-Roots of Integers

On the Lengths of the Periods of the Continued Fractions of Square-Roots of Integers Abstract. Let L = L(d) be the length of the period of the regulär continued fraction expansion of j/J where the positive integer d is not a perfect square. The first upper estimate for L was found by Lagrange and the essentially sharpest result is by Podsypanin stating that L = O (j/d log d/f2) where d/f2 is square-free. We show that for "most" /'s the stronger Statement L (d) = Oty^d) holds; more precisely, we prove the following: Let Kbe a given large number. Then there exists a natural number NQ -- NQ (K) such that for any N > N0 we have L(d) < e2K]fdfor dsatisfying N+l <d< 2N with at most cN/K2 exceptions where c denotes an absolute constant. The proof is based on Chebyshev's classical inequality in probability theory, on certain well-known relations in prime-number theory, and on Selberg's upper bound sieve. 1980 Mathematics Subject Classification (1985 Revision): 11A55, 11N35, 11K50. Let d be a positive integer that is not a perfect square and let (1) [ 0 ;* ,*2,···] = 1/5 be the regulär continued fraction expansion of j/rf. A classical result of Lagrange (see, for instance, Perron [3], p. 63) states that http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Forum Mathematicum de Gruyter

On the Lengths of the Periods of the Continued Fractions of Square-Roots of Integers

Forum Mathematicum , Volume 2 (2) – Jan 1, 1990

Loading next page...
 
/lp/de-gruyter/on-the-lengths-of-the-periods-of-the-continued-fractions-of-square-fxC10OjTcV

References

References for this paper are not available at this time. We will be adding them shortly, thank you for your patience.

Publisher
de Gruyter
Copyright
Copyright © 2009 Walter de Gruyter
ISSN
0933-7741
eISSN
1435-5337
DOI
10.1515/form.1990.2.119
Publisher site
See Article on Publisher Site

Abstract

Abstract. Let L = L(d) be the length of the period of the regulär continued fraction expansion of j/J where the positive integer d is not a perfect square. The first upper estimate for L was found by Lagrange and the essentially sharpest result is by Podsypanin stating that L = O (j/d log d/f2) where d/f2 is square-free. We show that for "most" /'s the stronger Statement L (d) = Oty^d) holds; more precisely, we prove the following: Let Kbe a given large number. Then there exists a natural number NQ -- NQ (K) such that for any N > N0 we have L(d) < e2K]fdfor dsatisfying N+l <d< 2N with at most cN/K2 exceptions where c denotes an absolute constant. The proof is based on Chebyshev's classical inequality in probability theory, on certain well-known relations in prime-number theory, and on Selberg's upper bound sieve. 1980 Mathematics Subject Classification (1985 Revision): 11A55, 11N35, 11K50. Let d be a positive integer that is not a perfect square and let (1) [ 0 ;* ,*2,···] = 1/5 be the regulär continued fraction expansion of j/rf. A classical result of Lagrange (see, for instance, Perron [3], p. 63) states that

Journal

Forum Mathematicumde Gruyter

Published: Jan 1, 1990

There are no references for this article.