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.
An expression that can be expressed in the form:
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:
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]
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:
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:
Input: continued_fraction((10/7)) Output: [1, 2, 3] Explanation:
Input: list(continued_fraction_convergents([0, 2, 1, 2])) Output: [0, 1/2, 1/3, 3/8] Explanation:
Input: continued_fraction_reduce([1, 2, 3, 4, 5]) Output: 225/157 Explanation:
Code: To convert a fraction into Continued Fraction representation
[1, 2, 3]
Code 2: To convert a Continued Fraction into fraction.
Code 3: To get a list of convergents from a Continued fraction.
[0, 1/2, 1/3, 3/8]
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