|
|
| Line 1: |
Line 1: |
| In [[number theory]], the '''continued fraction factorization method''' ('''CFRAC''') is an [[integer factorization]] [[algorithm]]. It is a general-purpose algorithm, meaning that it is suitable for factoring any integer ''n'', not depending on special form or properties. It was described by [[Derrick Henry Lehmer|D. H. Lehmer]] and [[R. E. Powers]] in 1931,<ref>{{cite journal|last = Lehmer|first = D.H.|coauthors = Powers, R.E.|title = On Factoring Large Numbers|journal = Bulletin of the American Mathematical Society|volume = 37|year = 1931|issue = 10|pages = 770–776|doi = 10.1090/S0002-9904-1931-05271-X}}</ref> and developed as a computer algorithm by Michael A. Morrison and [[John Brillhart]] in 1975.<ref>{{cite journal|last = Morrison|first = Michael A.|coauthors = Brillhart, John|title = A Method of Factoring and the Factorization of ''F''<sub>7</sub>|journal = Mathematics of Computation|volume = 29|issue = 129| pages = 183–205|accessdate = 2007-05-13|date=January 1975|doi = 10.2307/2005475|jstor = 2005475|publisher = American Mathematical Society}}</ref>
| | Hi, everybody! My name is Irish. <br>It is a little about myself: I live in Australia, my city of Mount Coolon. <br>It's called often Eastern or cultural capital of QLD. I've married 1 years ago.<br>I have two children - a son (Krystal) and the daughter (Savannah). We all like Rock stacking.<br><br>My webpage; [http://support.file1.com/entries/35126614-Hostgator-Application-X-Httpd-Php5 http://Www.hostgator1centcoupon.info/] |
| | |
| The continued fraction method is based on [[Dixon's factorization method]]. It uses [[Convergent (continued fraction)|convergents]] in the [[continued fraction|regular continued fraction expansion]] of
| |
| :<math>\sqrt{kn},\qquad k\in\mathbb{Z^+}</math>.
| |
| Since this is a [[quadratic irrational]], the continued fraction must be [[periodic continued fraction|periodic]] (unless ''n'' is square, in which case the factorization is obvious).
| |
| | |
| It has a time complexity of <math>O\left(e^{\sqrt{2\log n \log\log n}}\right)=L_n\left[1/2,\sqrt{2}\right]</math>, in the [[Big O notation|O]] and [[L-notation|L]] notations.<ref>{{Cite news|last=Pomerance|first=Carl|author-link=Carl Pomerance|title=A Tale of Two Sieves|date=December 1996|periodical=Notices of the AMS|pages=1473–1485|volume=43|issue=12|url=http://www.ams.org/notices/199612/pomerance.pdf|postscript=<!--None-->}}</ref>
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| {{number theoretic algorithms}}
| |
| | |
| [[Category:Integer factorization algorithms]]
| |
| | |
| | |
| {{Numtheory-stub}}
| |
Latest revision as of 18:31, 7 January 2015
Hi, everybody! My name is Irish.
It is a little about myself: I live in Australia, my city of Mount Coolon.
It's called often Eastern or cultural capital of QLD. I've married 1 years ago.
I have two children - a son (Krystal) and the daughter (Savannah). We all like Rock stacking.
My webpage; http://Www.hostgator1centcoupon.info/