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Continuing this process until every factor is prime is called prime factorization; the result is always unique up to the order of the factors by the prime factorization theorem. To factorize a small integer n using mental or pen-and-paper arithmetic, the simplest method is trial division : checking if the number is divisible by prime numbers 2 ...
In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10 100. Heuristically, its complexity for factoring an integer n (consisting of ⌊log 2 n ⌋ + 1 bits) is of the form
Shanks' square forms factorization is a method for integer factorization devised by Daniel Shanks as an improvement on Fermat's factorization method. The success of Fermat's method depends on finding integers x {\displaystyle x} and y {\displaystyle y} such that x 2 − y 2 = N {\displaystyle x^{2}-y^{2}=N} , where N {\displaystyle N} is the ...
In number theory, Dixon's factorization method (also Dixon's random squares method [1] or Dixon's algorithm) is a general-purpose integer factorization algorithm; it is the prototypical factor base method. Unlike for other factor base methods, its run-time bound comes with a rigorous proof that does not rely on conjectures about the smoothness ...
In computational number theory, Williams's p + 1 algorithm is an integer factorization algorithm, one of the family of algebraic-group factorisation algorithms. It was invented by Hugh C. Williams in 1982. It works well if the number N to be factored contains one or more prime factors p such that p + 1 is smooth, i.e. p + 1
In mathematics, factorization (or factorisation, see English spelling differences) or factoring consists of writing a number or another mathematical object as a product of several factors, usually smaller or simpler objects of the same kind. For example, 3 × 5 is an integer factorization of 15, and (x – 2)(x + 2) is a polynomial ...
Pollard's rho algorithm is an algorithm for integer factorization. It was invented by John Pollard in 1975. [ 1 ] It uses only a small amount of space, and its expected running time is proportional to the square root of the smallest prime factor of the composite number being factorized.
Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors; it is the simplest example of an algebraic-group factorisation algorithm.
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