Continued fraction factorization

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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,^[1] and developed as a computer algorithm by Michael A. Morrison and John Brillhart in 1975.^[2]

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^[3].

[edit] References

^ Lehmer, D.H.; Powers, R.E. (1931). "On Factoring Large Numbers". Bulletin of the American Mathematical Society 37: 770-776. doi:10.1090/S0002-9904-1931-05271-X.
^ Morrison, Michael A.; Brillhart, John (Jan 1975). "A Method of Factoring and the Factorization of F₇". Mathematics of Computation 29 (129): 183-205.
^ Pomerance, Carl (December 1996), “A Tale of Two Sieves”, Notices of the AMS 43 (12): 1473-1485, <http://www.ams.org/notices/199612/pomerance.pdf>

v • d • e Number-theoretic algorithms

Primality tests	AKS · APR · Ballie-PSW · ECPP · Fermat · Lucas–Lehmer · Lucas–Lehmer (Mersenne numbers) · Lucas–Lehmer–Riesel · Proth's theorem · Pépin's · Solovay-Strassen · Miller-Rabin · Trial division

Sieving algorithms	Sieve of Atkin · Sieve of Eratosthenes · Sieve of Sundaram · Wheel factorization

Integer factorization algorithms	CFRAC · Dixon's · ECM · Euler's · Pollard's rho · P - 1 · P + 1 · QS · GNFS · SNFS · rational sieve · Fermat's · Shanks' square forms · Trial division · Shor's algorithm

Other algorithms	Ancient Egyptian multiplication · Aryabhata · Binary GCD · Chakravala · Euclidean · Extended Euclidean · integer relation algorithm · integer square root · Modular exponentiation · Shanks-Tonelli

Italics indicate that algorithm is for numbers of special forms; bold indicates deterministic algorithm for primality tests.