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,[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

kn,k∈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(e2log⁡nlog⁡log⁡n)=Ln[1/2,2], in the O and L notations.[3]

## References

1. ↑ Lehmer, D.H.; Powers, R.E. (1931). "On Factoring Large Numbers". Bulletin of the American Mathematical Society 37 (10): 770–776. doi:10.1090/S0002-9904-1931-05271-X.
2. ↑ Morrison, Michael A.; Brillhart, John (January 1975). "A Method of Factoring and the Factorization of F7". Mathematics of Computation (American Mathematical Society) 29 (129): 183–205. doi:10.2307/2005475. https://www.ams.org/journals/mcom/1975-29-129/S0025-5718-1975-0371800-5/.
3. ↑ Pomerance, Carl (December 1996). "A Tale of Two Sieves". Notices of the AMS 43 (12): pp. 1473–1485. https://www.ams.org/notices/199612/pomerance.pdf.

## Further reading

* Samuel S. Wagstaff, Jr. (2013). The Joy of Factoring. Providence, RI: American Mathematical Society. pp. 143–171. ISBN 978-1-4704-1048-3. https://www.ams.org/bookpages/stml-68.

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Number-theoretic algorithms

Primality tests|

* AKS
* APR
* Baillie–PSW
* Elliptic curve
* Pocklington
* Fermat
* Lucas
* Lucas–Lehmer
* Lucas–Lehmer–Riesel
* Proth's theorem
* Pépin's
* Quadratic Frobenius
* Solovay–Strassen
* Miller–Rabin

Prime-generating|

* Sieve of Atkin
* Sieve of Eratosthenes
* Sieve of Sundaram
* Wheel factorization

Integer factorization|

* Continued fraction (CFRAC)
* Dixon's
* Lenstra elliptic curve (ECM)
* Euler's
* Pollard's rho
* p − 1
* p \+ 1
* Quadratic sieve (QS)
* General number field sieve (GNFS)
* Special number field sieve (SNFS)
* Rational sieve
* Fermat's
* Shanks's square forms
* Trial division
* Shor's

Multiplication|

* Ancient Egyptian
* Long
* Karatsuba
* Toom–Cook
* Schönhage–Strassen
* Fürer's

Euclidean division|

* Binary
* Chunking
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* Goldschmidt
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* Long
* Short
* SRT

Discrete logarithm|

* Baby-step giant-step
* Pollard rho
* Pollard kangaroo
* Pohlig–Hellman
* Index calculus
* Function field sieve

Greatest common divisor|

* Binary
* Euclidean
* Extended Euclidean
* Lehmer's

Modular square root|

* Cipolla
* Pocklington's
* Tonelli–Shanks
* Berlekamp

Other algorithms|

* Chakravala
* Cornacchia
* Exponentiation by squaring
* Integer square root
* LLL
* Modular exponentiation
* Montgomery reduction
* Schoof's

* Italics indicate that algorithm is for numbers of special forms

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