continued fraction factorization

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 D. H. Lehmer and R. E. Powers in 1931,{{cite journal|last = Lehmer|first = D.H.|author2=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|doi-access = free}} and developed as a computer algorithm by Michael A. Morrison and John Brillhart in 1975.{{cite journal|last = Morrison|first = Michael A.|author2=Brillhart, John|title = A Method of Factoring and the Factorization of F₇|journal = Mathematics of Computation|url = https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0371800-5/|volume = 29|issue = 129| pages = 183–205|date=January 1975|doi = 10.2307/2005475|jstor = 2005475|publisher = American Mathematical Society}}

The continued fraction method is based on Dixon's factorization method. It uses convergents in the regular continued fraction expansion of

: $\sqrt{kn},\qquad k\in\mathbb{Z^+}$ .

Since this is a quadratic irrational, the continued fraction must be periodic (unless n is square, in which case the factorization is obvious).

It has a time complexity of $O\left(e^{\sqrt{2\log n \log\log n}}\right)=L_n\left[1/2,\sqrt{2}\right]$ , in the O and L notations.{{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=https://www.ams.org/notices/199612/pomerance.pdf}}

continued fraction factorization

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Further reading