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]
Attention geek! Strengthen your foundations with the Python Programming Foundation Course and learn the basics.
To begin with, your interview preparations Enhance your Data Structures concepts with the Python DS Course.