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Continued Fraction Factorization algorithm
• Last Updated : 02 Sep, 2020

The continued fraction factorization method (CFRAC) is a general-purpose factorization algorithm valid for integers. It calculates factors of a given integer number without considering its unique properties. It has a sub-exponential running time.  It was first described in 1931 by  D. H. Lehmer and R. E. Powers and later in 1975 were developed into a computer algorithm by Michael A. Morrison and John Brillhart.

Continued Fraction:
An expression that can be expressed in the form:

(1)

is called a Continued Fraction, where ai and bi are either real or complex values for all i > = 0. When all the values of bi‘s are 1, then it is called a simple continued fraction.

A Simple Continued Fraction can be denoted as:

(2)

where Ck= [a0; a1, a2, …, an] for k<=n is the k-th convergent of the Simple Continued Fraction.
An Infinite Continued Fraction [a0; a1, a2, …, ak, …] is defined as a limit of the convergents Ck=[a0; a1, a2, …, an]

Algorithm:
This algorithm uses residues produced in the Continued Fraction of (mn)1/2 for some m to produce a square number.

This algorithm solves the mathematical equation:

(3)

this equation is solved by calculating the value of m such that m2 (mod(n)) has the minimum upperbound.

• CFRAC algorithm has a time complexity of:

(4)

Example 1:


Input: continued_fraction((10/7))
Output: [1, 2, 3]

Explanation:


(5)

Example 2:

Input:  list(continued_fraction_convergents([0, 2, 1, 2]))
Output: [0, 1/2, 1/3, 3/8]
Explanation:


(6)

Example 3:


Input: continued_fraction_reduce([1, 2, 3, 4, 5])
Output: 225/157
Explanation:


(7)

Implementation:
Code: To convert a fraction into Continued Fraction representation

 #using sympy module  from sympy.ntheory.continued_fraction import continued_fraction  from sympy import sqrt  #calling continued_fraction method  continued_fraction(10/7)

Output:

[1, 2, 3]


Code 2: To convert a Continued Fraction into fraction.

 #using sympy module  from sympy.ntheory.continued_fraction import continued_fraction_reduce      #calling continued_fraction_reduce method  continued_fraction_reduce([1, 2, 3, 4, 5])

Output:

225/157


Code 3: To get a list of convergents from a Continued fraction.

 # using sympy module  from sympy.core import Rational, pi  from sympy import S  from sympy.ntheory.continued_fraction import continued_fraction_convergents, continued_fraction_iterator        # calling continued_fraction_convergents method and   # passing it as a parameter to a list  list(continued_fraction_convergents([0, 2, 1, 2]))

Output:

[0, 1/2, 1/3, 3/8]


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